You can't go wrong as long as you leave some leaves for photosynthesis. I would, for starters, cut off the growth coming from the base and then remove the awkwardly angled branches.
The aerial roots could be a result of not watering or a need for re-potting, hard to say which. I would cut it back with discretion, water it, and wait 'til winter to re-pot.
Also, if you leave it in full sun, it will turn all yellow and stop growing. Find a semi-shady spot.

I agree with hanzrobo. Get it growing well before you really go at it, although you can definitely get rid of some of those straggly bits.

Why haven't you watered it?!?!

How long has it looked like this? I would have watered at least twice in the month you've
had it. If I knew it had a healthy root-system, I would have watered three times at least.

Josh

Jeni,

Does it have any roots at all? If not, I would mist it every 2-3 days. read some threads on rooting jade stumps/cuttings.

Inna

Minus the aerial roots and a few of those scraggly/dangly bits that dropped away, the jade looks similar to how I received it. I haven't watered it because whenever I check it, the leaves feel firm. I'm worried too, that there's not enough of a root system yet to support a thorough watering. When I first received it, and unpacked it, the base was caked in the old potting soil. I couldn't get all of it off, but I removed as much as I could. I was really concerned that I didn't see any fine feeder roots, or anything I'd classify as roots (even before I cleaned it off), so thought it would be best that I waited until I saw new growth before I watered. How far off base am I?

Jeni

At this point, the roots it did have could have dried up and died, leaving it in even worse shape.
That said, it's in a gritty mix, so I would water it thoroughly and check back in a week.

Josh

If it's been in new soil for a month, it has roots. Jade cuttings of that size don't take long at all. It only needs the tiniest beginnings of roots to absorb water. I hope you have watered it thoroughly since starting this thread.

You said you checked the leaves and they felt firm. When C. ovata is thirsty and dry, it's leaves will stay firm. In a pinch, it will absorb the water from the lowest leaves on each branch until they are dried up. It will send out aerial roots to pull moisture from the air instead of using up all its leaves. Also, a good sized trunk holds quite a bit of sustenance for the plant. A trunk with or without leaves can sit for many months without being planted, especially during Spring/Summer when it's not growing very much.

I guess I still have a lot to learn... Thank you all for helping me! I watered it thoroughly this morning. I'll wait to prune until I see new growth or at least for another couple of weeks.

Jeni