As well as an Audio Interface you will also need to buy a microphone. It is possible to plug in an electric or acoustic guitar right into an Audio Interface but it generally sounds rubbish and should be avoided.
Dynamic vs Condenser
There are two main types of microphone: dynamic and condenser. And you will probably end up needing one or more of each type, they have different strengths and weaknesses.
Condenser mics are the most common type of studio microphone. They capture a wider frequency response than dynamic mics, and have a better transient response (the ability to capture faster moving sound waves). They generally require Phantom Power (+48v) to be supplied by the mixing desk, audio interface or microphone pre-amp. Condenser microphones are considerably more expensive than dynamics, and cheap ones generally do not sound good and are best avoided. Wiki has some info on how condenser mics work.
They come in two types: Large Diaphragm and Small Diaphragm Large Diaphragm (Neumann U87, AKG 414, Rode NT2) are used for vocals, drum overheads and guitar bodies and can have a big or fat deep sound. They are sensitive to volume change and not suitable to record loud sounds (like electric guitar) and you will need to use a pop shield to stop bursts of air (plosives, like "p" and sibilance like "shh") causing distortion. Small Diaphragm (Neumann KM184, Rode NT5) or "pencil mics" are better for guitar necks, close micing strings, drums and more, they have a better transient response and are often used in pairs to create a stereo spread.
My favorite mic for acoustic guitar is a medium diaphragm: a Neumann KM86, which is an old mic that is no longer made, but to my ears it sounds FAR better than any other mic or combination of mics on acoustic guitar and is worth searching out if you are wanting the best acoustic guitar sound possible, but they are not cheap (£1000+).
The classic vocal condenser microphone is the Neumann U87, which sound great, but are quite expensive (£2000, $3500). You don't have to spend that but there is a certain clarity that you find in Neumann mics that others just can't seem to capture. I used a Rode NT2-A and got some great results, in general I have found the Rode brand to make excellent quality microphones that are very good value (actually I just had the NT-2, the older version)...
Valve condenser mics are also available at considerable extra cost, and add a real warmth to the sound.
When choosing a vocal mic it is important to try out a whole heap, because sometimes the most expensive won't sound the best for you. You need to borrow as many as you can from a store and then try them all out - or go book to a studio for a few hours and try all their vocal mics) and see which one suits you. Don't miss out that step - try out loads and find the one that works FOR YOU!!
There is not a massive difference between medium and very expensive mics. The more expensive mics will certainly sound a little better, have better definition - but is it worth it for you? The Neumann KM184 and Rode NT5 are essentially the same model but the Neumann is more than 4 times the cost. The Neumann is better IMHO, but if you are starting out you would be crazy to spend so much more for such little improvement in sound, they are VERY similar.
Dynamic mics require no "phantom power" to be sent from the Audio Interface and the most common type of microphone for live use. They are generally tougher and suited to live use and have very few parts. Wikipedia has quite a lot on how dynamic microphones work too.
Probably the most common is the Shure SM57, a fantastic microphone for electric guitars, drums and many other applications and is not expensive (£60, $90). You will probably find many of these mics in pretty much any pro studio, and it's unlikely you will ever see a live show that doesn't use at least one. Some people even use them for vocals if they want a little bit of a crunchy sound (Mick Jagger likes them apparently).
The Shure SM58 is another common model - mostly used for live vocals, not so much in the studio. There are many hundreds of other dynamic mics, but the SM57 is the standard, just get one!!
Just as a side note there is also a type of microphone that I like a lot for guitar called a ribbon mic. They seem to take sharp edges off sounds, and I loved the Royer 121 for recording electric guitar. Took nasty sounds away and made it sing. I have T-Bone cheap ribbon mic too - which I have used a few times, but it's not the same league as the Royer.
Important note: DO NOT USE PHANTOM POWER WITH RIBBON MICS - you will probably cause them a lot of harm, especially if they are vintage.
So what mic should you get?
So your choice will come down to budget. Ideally I think you will need one dynamic and one condenser at least. You can probably find cheaper if you look around, but you'll probably want to improve your sound after not too long - and I always think... buy cheap, buy twice.
Look second hand too, you can save a lot and often old stuff sounds better!! Also check out Madooma (great vintage mics and excellent customer service IMO).
So here is my rough guide, but it is very subjective - lots of people like different things, so it's your call, ask around your friends and try some out if you can and see what works for you and how much money you have to spend.
Cheap as possible
Large (Behringer C-1) + Dynamic (M-Audio Soundcheck Dynamic)
Cheap but usable for a while...
Large (Rode NT1) + Dynamic (Shure 57)
Large (Rode NT2 or Neumann TLM103) + Dynamic (Shure SM57)
Large (Neumann U87) + Dynamic (Shure SM57) + Pencil (Rode NT5 or Neumann KM184)
I've got too much money...
Large (Vintage Neumann U87) + Medium (Vintage Neumann KM86) + Small (Neumann KM184) + Dynamic (Shure SM7) + Ribbon (Royer 121).
Hope that helps you choose, J.