No, you do not want a MIDI interface (to do what you are talking about doing anyway). A MIDI interface doesn't record, it is a way for you to get note data (midi data) into your computer.

You want an Audio Interface. Something like this....
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I only linked to that one in particular because I am familiar with it and I know that it is relatively easy to set up to muli-track (record multiple instruments or, in the case of a drum set, record different Mics at once with one instrument/mic per track). But there are tons more.... some more expensive, some less... it just depends on what you want to do.

EDIT: I'm assuming that you are looking to record, mix and edit on a computer....

Ok awesome thanks

Would you be able to recommend one with sliders for a more tactile feel?

are you willing to spend $800+ ?

ha no probably not right now
But what do you have in mind?

Something along the lines of this....

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There are some other mixers that connect to a computer via firewire... but I don't know much about them or how easy they are to get set up to multitrack.