I'm using 2 Delta 1010LTs synced together (love them). So, here's my 2 cents....
External mic preamps and EQs:
Advantages: Usually higher quality, highly tweakable, add more as you need them.
Disadvantages: Expensive, limited routing options without a patchbay, usually rackable, so you have to go to the rack to tweak the settings.
Mixer:
Advantages: A decent mixer will have quality mic preamps, a decent EQ, and good routing options for effects, subgroups, and insert channels, and it's really easy to control everything from a central console (faders, switches, etc.)
Disadvantages: Unless you spend a *lot* of money, a mixer's preamps and EQs will not be as good as stand-alone preamps and EQs. If you run out of channels, you have to buy a new mixer. Pro-levl mixers allow you too add channels as strip modules, but not the budget ones we're shooting for.
I've opted for a combination. I have an older Ramsa mixer which sounds pretty good when I need a lot of channels for preamp, EQ and routing. I have 2 stand-alone preamps when I only need 2 channels or for critical recording (vocals, overheads, acoustic guitar, etc.).
Many smaller project studios value Mackie and Soundcraft mixers for their mic preamps and EQs at budget prices.
Whatever you get, make sure it has enough input/output for you and phantom power for condenser mics.
Hope that's helpful!
Oh, and don't forget that your DAW software (n-Track, Cubase, Cakewalk, Reaktor, etc.) must be capable of recording as many simultaneous tracks as you're going to need.
-Rick