Of course my first vote is QH, but that's my thing :) I agree with other posters, don't worry about breed. Judge each individual horse, make a list of what you want, things that are deal breakers -for me my big deal breaker is cribbing, I like my fence too much to be replacing posts for a cribber or wind sucker not to mention the teeth problems & other health issues that can stem from it.
I agree with looking for something with some age on it, but don't forego a thorough PPE by a reputable vet. Ask the current owner to give their vet permission to release full medical records for your viewing, if they aren't willing RUN the other way.
I always tell folks I help find beginner horses, NEVER ride any horse that the owner won't ride first.
It's always a good idea to take someone knowledgable with you, so you have eyes on the ground to watch for any signs of unsoundness while being lunged or ridden.
Take your time, don't rush. The last beginner horse I found for a student was the 17th horse we looked at.
I am on the fence with the age thing - I've seen 12 year olds better experienced and better mannered than some 17 year old's or 20 year olds.
I wouldn't put a begginer on Nelson, that's for sure - he'd take full advantage of them shouting "WOO HOO! FREE RIDE!" - and he's 22!
It depends on the exerience level of the indavidual horse. Look for a good beginner mount, regardless of age and breed. You may find a 13 year old who is perfect for you - I'd hate to see you dismiss the horse, just because the age isn't over 16.
I agree with all the above, go by temperament and take your time, don't settle on the first horse that seems calm! Personally, I'm a big fan of draft crosses because they tend to be very docile and calm-mannered, but I've met thoroughbreds who were more sane then my own draft cross sometimes!
If you're ok with an older horse, maybe consider looking for retired lesson horses? I know my old barn gives away the older lesson horses once they get to be about 17-20 or when daily 1-2hours riding per day seems to be too much for them. They are either donated to handicapped riding so they work as much but mainly walking, or go to people's backyards that are looking for laid-back, infrequent rides. These horses were made for little kids to ride and are usually bomb proof so they'd probably be perfect to learn on.