Now it depends on how tall and heavy you are, there are lots of (mainly ladies) who think that they NEED to ride a draft and that is what they become comfortable with.
In fact because drafts were designed to pull rather than carry they are not always the best bet for us fluffy people.
Again, depending on if you really are a heavier rider, I would be looking firstly for great conformation, you just can't hide any problems a horse has if you are going to be riding long distance, especially if weight carrying, goes without saying that great hooves are also required.
I'm thinking fairly short backed and well sprung barrel, it makes carrying easier. Oh and a back that you can fit a 'standard' saddle on, not that there is any such thing, but ask if they have had any problem with saddle fit.
Don't be hung up on breeds, you are looking for a type, but, again depending on your height and weight, an Arab would be an obvius choice for what you want to do, or a mutt like my Appy, here we go, 5' 9" and at that time 240 pounds, on a 15. 1 hh Appy cross, who can go all day at whatever speed you want.