Most of Wildflowers was recorded in Mike Campbell's home studio on a Soundcraft 1600 that had a 12 channel Neve 1072 sidecar. Tom and Mike recorded most of it themselves, then Rick and an engineer (either Richard Dodd, or Jim Scott, I can't remember) was brought in for overdubs and mixing. Lots of what you hear was originally intended to be a 'scratch track', but the initial recordings were so good, and so full of energy, they were kept in the final mix. Some vocals on this album, recorded as scratch tracks, are actually an SM-57. Others are neuman mics, and it sounds to me, like a Philips C12, but I know its likely a vintage neuman.
Tom Petty usually records to Ampex 456 2" tape.
As for the drums, they probably spent a good week or two just picking out a set at one of LA's drum houses.
You can see on some petty tracks something called "multiband compression". Compression, as has been already stated, takes loud sounds, and makes them quieter, and quiet sounds, and makes them louder, for an overall more consistent volume level.
While there is certainly compression on various tracks of in the mix, multiband compression is typically used on the entire mix - that is, the stereo two-track mix, right before it hits the CD. (Sometimes the mix engineer does this, sometimes the mastering engineer does it.)
A multiband compressor is essentially a three or four band crossover (takes audio in, separates it into 'low', 'mid' and 'high' bands - most speaker systems use these to separate audio to various speakers), connected to three or four compressors. This way, if there's a loud bass 'thud', the vocal (midrange) doesn't get pulled down.
Good multiband compression can make nearly anything sound great - well, compressed.
I listen to a lot of music in iTunes, and on the analyzer, I can see the multiband compression. Just look for two or three distinctive groups of side-by-side bands that keep rising and falling in sync with eachother. There's usually one from about 750hz to 8khz.
Many engineers and producers criticize the use of compression, and especially multiband compression, sighting lower dynamic range, and the belief that the mix engineer is allowing a piece of equipment too much creative input. In fact, I believe that Wildflowers is the most natural sounding, compressed album ever. The creative use of compression is great, and this is a great example of it. Not too much, but just enough.
Tom Petty's sound also comes from good technique, and great musicians skill.