What do I need to consider when buying amp and sub

You can hook up a sub a few different ways. One is to run speaker cable to the sub, then from the sub to the mains speakers. For this your sub must have 2 sets of speaker cable connectors - one set in, one out. This allows you to use the sub's crossover so you can select a lower limit for your main speakers and upper limit for the sub. Not ideal since you have to run 2 sets of speaker cable, and go through the sub's crossover.

The most common way is to use 'pre-outs' from your integrated, using a y connector and single rca into your sub. Using this method you can't use the crossover in the sub, except to set the sub's upper limit. If you integrated does not have pre-outs or tape out, obviously this method won't work.

The third method, which is used by REL and MJ Acoustics (only as far as I know), is a high level Neutrix cable that connects to your speaker terminals on the amp, and has a single connector to the sub.

Hope this helps.

The JL subs don't take speaker level inputs-line level only with balanced XLR and SE RCA.

This is helpful, but it creates more questions. What is a line level connection and what are xlr and se? Are those like options 2 or 3 above?

Velodyne subs have the high level speaker input. Run 2 sets of speaker wire from your amp. One to your mains and one set to the left and right speaker inputs on sub. It is high level so it isnt gonna pull power, just obtain the signal. You can still use the sub Xover to blend into your system.
I use the vel with Focal 918s and set it just enough to add punch on the low end. works nice . Hope this helps.

I am pretty sure that I am going to get a JL Audio sub. What should I be looking for on an amplifier?

If you are going JL, you'll need an integrated amp with either pre/line level-outs or tape out.

Jeffatus - if your dealer sells the JL's then they will have all the answers. I myself went from RELs to a JL and it was the best move I've made regarding a sub. The JLs are best used via the XLR connection over the RCA connection and the better the cable the better the sound. I've tried 3 different XLR cables and finally settled on a Purist Provectus (dealer disclaimer) for the best results. The problem was the cable was as expensive as the sub! And don't forget you'll require decent power cord as well . Yes it's madness but in the end worth it.

If your main speakers have ports then most likely the best intergration will be achieved if your sub is ported