What problems will I face trying to edit these two different frame rates in AE?
I would highly suggest you DON'T use After Effects to edit. AE is terrible at editing, but it is great at doing motion graphics, compositing, visual effects, and animation. You should use Premiere Pro to edit.
Both After Effects and Premiere will work just fine with mixed frame rates. You should work with 30p compositions in AE and 30p sequences in Premiere and the footage will play back just fine. The bonus is that, if you want some slow-mo, you'll be able to get a much smoother result when you slow down the footage shot on the new camera.
Will I get a slow mo effect from the 60p?
No. As long as Premiere and AE interpret the clips as 60p.
Will I have to convert the 60p to 30p?
Yes, I do all my actual; editing in PP. But I do a lot of effects in Ae and will want to mix these rates. So you are saying to use a 60p timeline? Okay, but why?
So you are saying to use a 60p timeline? Okay, but why?
NO! Definitely not!
I said to use a 30p timeline.
The reason is that your footage will be more likely to feel the same if they're all the same frame rate. If you use a 60p timeline, the newer footage will feel a lot smoother than the old footage. Now, this would be useful if you wanted to have a flashback or dream sequence feel different. You could put out most of your video at 60p, but drop a comp (or sequence) at 24p in there for the dream/flashback. It'll have a more filmic feel that way and separated from the "reality" of the higher frame rate.
But, as I said, if you want everything to feel the same, work in 30p sequences and compostions.