All the above are excellent posts and to the point.
I will play "producer" for you for free right here and suggest a few guidelines to maximize your time.
I will reiterate that you should practice your asses off and have your intruments in tip top shape before going in. Have a professional intonate the guitars and have fairly new drum heads on the drums. Buy some new guitar cables and make sure that your amps are in good shape. If ANYTHING is a bit off, get it fixed, or borrow something that IS in good shape.
Drum tuning is what is going to determine most whether you get good results or not!!! Be realistic! Most drummers DON'T have any clue how to tune their drums to go to tape well. Sorry, I have to say this because I have seen it far too many times. I am working with a drummer right now who recorded and toured with a major label signed band before joining this band and when I started working with them I had to go through and start tuning his drums. He was amazed how much better they sounded and so was the band. THIS IS A GUY THAT WAS ON A MAJOR LABEL!!! But when he was with the signed band, he had tech's who dealt with that stuff. Imagine that!!!
Anyway, I suggest that unless your drummer really knows how to tune drums (a rarity!) that he get a crash course in doing so. The studio engineer may or may not know how to tune drums and if he does, you are going to pay studio time for him to do so. Come in with the drums pretty close and have someone there that knows how to dial them in quickly and all will be well. I cannot stress enough how a great drum sound on tape will improve the overall quality of your recording. Usually, you would spend 10 hours dialing in drums in a major recording project. You don't have this option, so you need to come in with the best you can offer.
Okay, enough on drums....
Bass. MAKE SURE the bass in intonated. MAKE SURE that your bass player has good control with his pick or plucking fingers. MAKE SURE that the bass is capable of creating a sound that is appropriate to your bands sound. If you have all that going for you, getting the bass sound in the studio will only take less than 10 minutes!
Guitar. I have found that too many guitar players play with too much gain on their sound, and with too little midrange. If you cannot hear EVERY NOTE of the guitar while you are practicing live, you probably have a tone that is not very good. That is a fact that I have proven to guitar player and guitar player in the studio. I don't know your style or what kind of tone you are using right now, but if you remember that to get a good sound on tape that you usually need far less gain than you think, and having good midrange in the sound will give the best results, you will have little trouble in the studio. MAKE SURE that the guitar intonates well with the bass!!! Have the same guy intonate both intruments and make sure to play a little for the guy before he intonates so he can get an idea of how you play. This will help him do his work much better. Trust me on this one. If the luthier tries to tell you that your technique in playing won't guide how he works on the guitar and bass in setting it up, go somewhere else!!! He obviously doesn't understand the intrument well enough to be worth the price.
Vocals. If you have back up vocals, PRACTICE them!!! I find most backup singers are out of tune when they come into the studio. RECORD yourselves practicing vocals! Tune the vocal way up in the mix and LISTEN. If they make you cringe, something is wrong....

Fix that BEFORE you try recording. For lead vocals, if you find that the meters on your br 8 are all over the place while you are tracking, you may need to start making some adjustments in how you present the singing to a microphone. Most inexperienced singers are overly dynamic.
For the session. Get a LOT of sleep the night before. 10 hours is going to be gruling! Eat pasta the night before because the energy it produces will pay off the next day. This is a trick distance runners use. Load up on pasta the day before. During the session, bring a friend along who will go get food or drinks during the day for you so none of the band members need to leave the studio to do this. Make sure to pack up all your gear and maybe even pack it in a van the night before so you don't have to exert that energy the day that you will be trying to track music! PREPARE as much as possible so your day goes smooth.
When you get to the studio, don't spend a bunch of time Ooooooing and Ahhhhhhing over the gear and rooms and stuff. Get you gear in and start setting up. I suggest that you record the drums bass and guitar at the same time. Ask the engineer what he feels is going to be the best setup in the room to do this. FOLLOW his advice. You don't have a bunch of time and if you present yourself as a professional to the engineer he will deal with you in a professional manner. At all costs, don't bother him with a whole slew of questions on small little things and come across as worrying about how stuff is going to turn out if you place your cable in one direction or another on the floor!

Sweating the engineer is on the verge of being insulting to him. He is a professional (I will assume...

) and probably knows what he is doing. Go about your business of setting up and getting ready to record and let him sweat the other stuff without interference. If you act like a dork, he will probably think you are a dork, you know what I mean?
Stay focused and don't do a lot of grab ass and fooling around in the studio. LISTEN to the engineer. If he suggests something, it would behove you to heed that advice. If this studio is in fact "up to date" as you mentioned, the engineer is probably pretty competent and knows how to get things done that will sound appropriate. If you guys are busy playing grab ass and talking insistently and arguing with him over sounds and what not he will probably become far less helpful in important things and will just sit around and wait for you to "okay" everything. If you are focused, well mannered, and trusting, he will work that much harder on your songs to achieve the best results.
It is okay to ask the engineer to suggest what you might change to help the sounds get better. If the engineer is experienced, he will be able to asses your talent and gear potential quite well and if you are cool and insistent about seeking his advice he will probably steer you in a good direction. LISTEN to what he has to say and ask him to clarify if you don't understand. Go about following his advice with an open mind even if you think it is crazy. He will probably impress you with his ideas more than you would think!
I suggest about 45 minutes to set up. You SHOULD be able to set up in 45 minutes! If you can't you are either lazy or have a setup that is too complicated.
I would spend about 1 or 2 hours on getting sound on the drums. The more the better. "Good enough" after 30 minutes is certainly NOT good enough. 2 hours is not enough time either but that is about all you can afford with only 10 hours total time to record and mix. Tell the engineer how much time you want to spend and make sure that the drummer stays focused and motivated during this whole time. Have a drum key ready. I forgot one thing above about drums. If the drummer has cymbals that are very low that are right over the toms you are going to have some problems in the studio. Cymbals that close to toms means that if you try to close mic the toms that the tome mics will pick up a lot of cymbal bleed. Also, it limits where the mic can be placed to mic the toms. I get the best drum sound froms drummers that play their cymbals at least 1' above any drum. Something to think about.
Spend about 15 minutes on getting a good bass sound. If your bass player as a good bass head with a direct output, bring it. I really wouldn't worry about bringing bass speakers as that requires a lot of time to set up and mic well. Go DI with a DI on the instrument and on from the bass amp. When the bass player is playing while working up a sound, don't have him playing funk bass riff unless he is going to be playing a funk song! God I hate that!!! When checking sounds, play YOUR songs. This way the engineer knows what is going to be played.
Guitar. I would recommend spending about 15-30 minutes on working up a good guitar sound. This is about all you can afford in time. The engineer may want to try a close mic and a room mic if he has a seperate room for the guitar amp and it is big enough for a room mic. Recording both is not a bad idea and the depth a room mic can offer cannot be replicated with reverb too well. If you are going to be going between a clean and dirty sound, it would be better to have two amps to do so, or two mics on the amp going to two different tracks. This will allow the engineer to have a track for the dirty sound and one for the clean sound and better level adjustment and eq can be applied. If you envision the dirty and clean parts overlapping during the transitions between them, you will need to track them seperately. Go for it if that is what you need to do. But, with so little time available, it might be better to try to avoid a lot of overdubbing. If you have a guitar solo, I would overdub that, especially if you want rythem guitar behind it.
Okay, your sound are now worked up. You have used between 2.5 and 3.5 hours to do so. Take a piss break and smoke a cigarette if that is what you do. DON'T get high! DON'T drink!! Drugs and quick recording do not mix. Stay FOCUSED!!!
I would recommend tracking three songs. Get in and play them. Spend about 10 minutes on getting the headphone mix good. You are now at between 3 and 4 hours and you are ready to track.
At this point, relax and play. Don't tighten up. Play the songs the way you rehearse them. Don't worry about making a mistake and you will be far less inclined to make one. If you DO make one on the bass or guitar, DON'T stop!!! If the drummer makes a big boo boo, STOP and start over.
If you cannot get the song in 3-5 takes, you didn't prepare. I WOULD not do any more than 5 takes. If you can afford the tape, record them all! If you think one of them sounded better than the others, go listen to it immediately and decide. ASK THE ENGINEER which he thought was best, but go with what you feel most comfortable with. If there was any gross bass or guitar mistakes, punch in and fix it. Move on.
If your songs are about 3 to 4 minutes, 3 takes of each song with one more listen to make sure you want to use a version, along with a few punch in's will take about around 1.5 hours to do. You now have used between 4.5 and 5 hours.
If you are going to overdub, do so. Again, 3-5 takes! If you can't get it right in that many times, you didn't prepare. Say another 30 minutes to overdub solos! You now have used 5 to 5.5 hours!
Get the vocalist in there. Three songs. Should take about 1 hour to track vocals. If there are back ups, record them along with the lead. Again, punch in on small mistakes.
You have now used between 6 and 6.5 hours.
Spend the rest of the time mixing. 3 songs, 3.5 to 4 hours to mix. I would spend the first two hours on the first song. The other two will go much quicker. During mix, work more on getting everything to gel together. Don't sweat a bunch of effects. Applying more than a reverb over the drums and possibly a tad on the guitar and a simple reverb and possibly a delay on the vocals starts getting very time consuming and you don't have time! Try to keep the reverbs short and ambient. Big lush reverbs will take much more time to dial in.
After you run your mix the way you "think" it shoud be, run one more with the vocals about 2 or 3 dB louder.
You at this point should have used up your 10 hours and should have a DAT or CDR of the three songs that sound pretty decent providing that all your source sound were good and everyone stayed focused.
You should inquire as to whether a one hour "lunch break" will count against your time. If it don't , take a hour break before you mix.
Ed