Number of Away Applications

It depends on a couple of things...first, consider how competitive the program is. It's no secret that programs often prefer rotators, so if it's a popular place, expect some competition. With that said, it's FAR easier to get the away you want than it is to get an interview. This can be a two-edged sword: you don't want a program letting you waste a month of your time rotating there if they're not even going to offer you an interview - and that does happen.

I would pick two places you really want, get the applications in early (through VSAS or whatever they use), and maybe have one to two back-ups. There have been rumors that getting accepted to do an away and then turning it down can burn bridges for interviews - I doubt that's true but I still think it's poor form to take a shot-gun approach to applying for aways (save that approach for your ERAS). Remember, there are other people out there stressing about getting aways as well.

One other piece of general advice about aways: know what you're getting. Call residents at the programs and ask questions up front. One good one is: how much input do residents have in the rank list? If you're a stellar applicant on paper this may not matter since you can impress both the residents and attendings. If you're a little weaker on paper, you want to be at a place that takes a lot of resident feedback on the rank list.

What GAdoc1982 said is good stuff. I have a few things to add, from my experience last year.

Number of rotators at a site matters. There are away sites with 50+ rotators a year, 10+ rotators per month - You're competing for interview offers first, then residency spots. At these places, not all rotators get an interview. You'll really have to shine.

The service you select or get assigned to matters. Some services are better than others when it comes to exposure to the residents or key faculty. On one my aways, an attending straight up told me the first day that their service/specialty had little pull with the people in charge. I still worked hard, learned a lot, and got a great letter, but didn't get much exposure to the people who help make the decisions. I think that's where talking to the residents beforehand would have been helpful.

I myself just went through this process and managed to get both aways I wanted. Basically, the earlier you apply the better. If you know what institution you want to do your aways at, find out when they start accepting applications and get yours in as close to that date as possible. Applying to institutions who don't use VSAS makes this significantly easier in some cases. And yes, there are some very good programs that don't take VSAS.
Not only that, but if the away requests something from you, like rec letters, or a personal statement of some kind, make sure you tailor it to the institution. Have letters from someone who is related to the program (career-wise, with research, location) or make your statement pertinent to why you want to go to that area.
And if you know someone connected to the program, or a resident in the program, their input is often invaluable.
That said, don't oversend applications for aways. If you end up turning down an away, you have basically burned a bridge for an interview. So try to get your first choices, when you hear back, move down your list if need be. But have everything ready to go to each institution so you can fire off the applications right when you know you need to.
Good luck.

Thanks for the great advice. I appreciate the input. Best of luck if any of you are matching tomorrow.