The device you're looking for is generally called an "audio interface", which is conceptually the same thing as a soundcard, but is usually oriented towards recording. There are a TON of different interfaces on the market aimed at different purposes and price points. You are wise to ask "what are the tradeoffs?" as opposed to "what's the best I can get at this price?"
The major things to worry about are:
Number and type of inputs
Assuming that you go with the mic/DI setup you described (and that is a fairly common straightforward kind of setup), you will need, at a minimum, one audio input for each instrument you plan to record simultaneously. So if you want to play with your whole band at once, you'll need one input for the guitar mic, three for the drum mics, and one more for the bass mic or DI. You will need an interface with 5 inputs. If you also want to do vocals concurrently, you'll need another input.
Most microphones use an XLR style connector, whereas most instruments and DIs use a 1/4" phone plug. Make sure that your interface has the correct type of inputs for what you want to plug into them. Many interfaces have "combo jacks" which can accept both 1/4" and XLR.
If you're using any condenser microphones, you'll generally need phantom power. Many interfaces have this, but you'll want to make sure that you can power your mics appropriately. Make sure that the phantom power won't mess up any other mics - you usually can't turn it on and off for individual inputs. Sometimes it's done in groups.
If you need the ability to monitor your signals through headphones before they reach the computer, make sure it's there (this is usually called "hardware direct monitoring")
Make sure it can connect to your computer - USB and Firewire are common connection methods, although you can also get PCI cards and such.
In general, this is the quality of the preamplifiers, although the rest of the electronics can have an effect as well. Most interfaces have preamps, and almost all mics require them, even when they're used on guitar amps. This is where you'll notice the largest gains in moving to a more expensive interface, and is generally what people are talking about when they say that a given interface has a particular "color" or "flavor" to it. It is, like everything in audio quality, very subjective.
You generally cannot escape latency in digital recording. The power of your computer will help, but some interfaces have more built-in latency than others. Dealing with this is a common problem and can get fairly technical. The best way to find out if a given interface will work is to try it, but failing that, looking for reviews by people on similar computers is the best method I've found to evaluate a given interface.
Stability and support
Terrible driver support will ruin even amazing hardware. My last interface was stable on Windows but would crash OSX once a week, due to its drivers (they showed up in the crash log). Check the reviews and see who's having issues. My experience has generally been that you get what you pay for here.
This rabbit hole can go as deep as you like, but these are the things I've found most prominent when I've gone interface shopping. Other, smaller concerns are things like what software it bundles, do you like how it looks/works, can you fit it in a bag if you travel a lot, or whether it has some nifty feature you want.