you can register with as many as you like though you may find some want to do home checks first. my advice is try to go to somewhere close by asyou will need to visit a few times even when you've found a potential match. Some rescues are strict about age because of known factors in dog. Others just think young kids and dogs don't mix.
To my knowledge the only time you will have problems getting a dog would be over xmas when most don't release dogs (because of chaos in homes!). It is best to visit out of peak times as you will find them quieter and more able to answer questions.
You don't need breed in mind - a cross can be as wonderful as a hugely pedigree dog! Think about your budget for the dog itself (all centes will ask for donation, ours is max of £250 which is a bargain) and the ongoing costs (not cheap: food, vet, insurance, kenneling, classes etc).
You will find lots of websites around that 'suggest' suitable dogs for types of families but you will recognise the type of dog you want: sociable, good with kids, unflappable, good with small furries or cats, generally healthy breed; and youwill probably come across dogs where you live. Ask everybody about their breed/cross and what he or she is like. Get a real feel for a dog this way and you will find it easier to choose one.
But in the end you will probably fall in love with one! Mine's a labradoodle (and I hasten to add we have had the breed fr donkeys' years before the celebs did!) and I'd not change, ever.