Welcome man!
I have done a bunch of hip hop/rap recording here in my studio. You are likely most importantly needing to focus on clear, recorded vocals on top of pre produced beats I assume? If so then below would be my personal suggestions for ya. Just my opinions. There will be many others with different ideas.
High end laptop or better, a well built PC tower. You can get more bang for buck with a PC. I'd say min of 16GB ram with SSD C: drive that runs your software. You will want to record to a hard drive separate from your C drive. You will have a bottleneck using the same drive and your performance will suffer. Video card is less of a worry recording audio, so a PC built for gaming may be way overkill. All you need is a fast processor, two drives and lots of ram for recording/production/effects.
-oh, and having two video monitors makes life in the DAW world way more efficient! I have three. Well, 6 including the two for my office PC, 3 for the DAW and one for the security system. I'm a real estate whore tho... lol
For your particular voice in a likely less than ideal space, I would recommend a Shure SM7B. Keep in mind a 'Cloud Lifter' may be needed to get decent gain from the mic. Klark Teknik CM-1 works fine on the cheap. Shure also now makes a SM7dB that has a preamp built in. Haven't used that version myself but reviews are good. You are more likely to find an SM7B used and get the CM-1 for less than the new version. Just a thought...
Maybe a Steinberg UR22 interface. I have two of the UR824's that have been super dependable for like a decade., The drivers are solid, and the preamps are quiet and high gain. Plus the UR22 comes with Cubase AI. That should be enough DAW for you to get started and see if you like it. Plus you would have me to contact directly to get you going.
Decent quality closed back headphones for tracking. Maybe open back for mixing. I use Sony MDR 7506 for tracking. I don't mix on headphones so don't have a suggestion there.
Good monitors are best to mix with in a well treated and ideal space, but depending on the room they are in, they can be tough to work with. Even in my well treated control room and ADAM A7x's, due to room modes, I have like no 80hz in my space due to a 'null' in frequency response. Subwoofer and countless hours 'learning' my monitors in here allow me to compensate correctly.
DAW software will be what works best for yourself. You will just have to try them and see. I use Cubase myself. You may only need the basic Cubase that comes with the UR22. Reaper seems popular and inexpensive. Studio One is quite similar to Cubase.
Most plugins (effects) supplied with a DAW will be sufficient to start out - until you know enough to know what else you need. Even after countless years of recording bands, I still use much of the included with DAW effects. Though there are boutique specialty plugs that I just can't live without now.
I would recommend for you 'Howard Benson Vocals' plugin. It will give you all of the basic vocal processing in one GUI that can get you started. Hell, I use it all the time. The reverb is kind of weak though. It has compression, saturation, de-essing, eq, doubler, widener, delay, reverb, limiter, lofi, all in your face to play with. Also a butt-load of presets to give you ideas and approaches. Compression and a bit of plate reverb is a good starting point for hip hop vocals. You want most of your lines up front/in your face. Don't get carried away with verbs man.
I would use as much as you can of your budget on room treatment. In a shed, you are going to have a hard time with 'isolation'. Best just dealing with sound that comes in and out. The SM7B will help with not hearing birds chirping from 2 blocks away like a condenser mic will get you. Your biggest issue in a small space is going to be room modes. You are at their mercy. Bass trapping in corners will help, but you will never get it perfect... More important to deal with immediately is first reflection points and flutter echos. This can be helped with DIY or purchased rockwool panels. 4" thick is your best bet.
Just say no to any foam product - Expensive and only work on high frequencies. In corners some use 6" rockwool panels or superchunks, but actually stacking rolls of pink stuff in corners is actually more efficient at trapping low end. It just depends upon how much space you have to work with. You are basically just chasing your tail treating low end in a 12'x12'x8' room...
What size and what type of construction is your shed?
Cheers!
