This might be the simplest answer on the planet but i actually have a bit of experience taking picture of shelter animals.
and the short answer is get a second person to help hold the dog or a cat and maybe bribe it with some treats while you shoot. Get anyone in the shelter to help hold the pets if possible. Someone there should be able to help.
In terms of a good first starter DSLR - we have a thread on this exact topic here: Good starter Digital SLR?
Hope it helps
Marko
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