When there's a section in a song or piece in which the tempo changes, how do you deal with this?
If it starts with a long note, I could just listen to the drums or something, but what do I do when it jumps straight to 16th notes in a whole different tempo?
Is it just a matter of playing and listening to it a lot, or is there a better way I'm missing?
It depends on if the tempos are linked in some way. A common tempo change is a dotted 8th becomes a normal 8th note or something like that. But sometimes the tempos aren't linked in any way, and you just kind of feel it. It's about connection.

How do you know what tempo to start playing the song in? When the tempo changes, it's the same thing. You just know how it should go. But yeah, a band can (and should) communicate - I mean, it's not just a group of musicians playing at the same time, it's a group of musicians playing together. That's how a band works. You need to listen to each other, but you can also look at each other to get clues of where the music is going. Use body language (for example nod your head or whatever). If you were playing acoustic instruments, you could breathe in tempo - this is what classical musicians do. But that doesn't really work in a loud rock band.
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Back in our fusion days we did a lot of tempo changes and odd time signatures. We usually lean on the drummer to lead us in with a transition measure but sometimes you just have to all go together and everyone needs to be there. That requires practice and counting out the changes.
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xD Often times I hold a note and stare at the drummer for a signal...which I know is not all that possible playing alone
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Lots of practice.

If you think about it, tempo can be thought of as the length of time between two notes. The first note of a tempo change will arrive at the expected moment, relative to the earlier tempo. As long as you hit the second note at the correct moment, you will be able to ride the tempo change successfully.

Concentrate hard on hitting that second note in tempo and listen to your band-mates. I usually look at the drummer and follow his movements to help me get it accurately.
What's the transition? Does it slow down/speed up, or is it a time sig change, like to 5/8 or something, or triplet time?

Often I find time changes simply involve switching into/out of triplets over whatever sig you are playing in, so it might crawl along almost like a 3/4, or it could start feeling like it's hauling ass, like a 6/8 feel, but either one can be played over a 4/4 pretty easily.

What's your genre? That makes a big difference as well.

If it's a metal band, for example, a section with the really sick fast double kick might just be triplet 16ths over the 4/4. Sounds insanely fast, but in actuality it's just "tri-pull-it, tri-pull-it.." and the snare still accents 2 and 4.

Anything Jazz oriented I'd look into the ethnic roots of the timing. Could be Latin or Zydeco, any number of different ethnic feels.

A good exercise I find is to try and work out different interpretations of songs on the radio, anywhere, really. If you're stuck at Wal-Mart (yuck!) see if you can figure out a Reggae feel of the crap on the speakers, tap it out on your shopping cart.

It all comes together eventually. You just have to work at it.

If the actual tempo changes, often it's a multiple of the previous tempo, like 90 to 120, so 30 extra beats/minute, that type of thing. There's more often then not a fairly simple math ratio behind it. If it's not directly half-time or double-time, I bet it's 33% faster or something.
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