There was an anecdote (like many Soviet-era anecdotes, hard to substantiate), that one of the top animal trainers in the Soviet space program went to a circus trainer and asked about using monkeys. When he found out how hard they were to deal with, they decided to go with dogs. And indeed the US program did have issues of monkeys that would occasionally get out of their harnesses in zero-G and not do things the way they were meant to be done. "The Americans are welcome to their flying monkeys," he allegedly remarked, "we're more partial to dogs."

Animals were explicitly seen as a prerequisite for later human work. The work with the dogs was scientific and not haphazard, part of a major biomedical research program. The two major candidates for this, for physiological and practical reasons, were apes and dogs. Dogs were easier to deal with than apes. Aside from their dexterity and relative rareness, apes are also more susceptible to human diseases, and were more excitable. They had been using dogs as test animals well before Laika was sent up — they put them into Soviet V-2s and other test rockets as well. There were at least 29 Soviet rocket dogs before Laika, starting as early as 1948. The decision to use dogs for the outer space program was finalized in 1950 by a commission of biomedical advisors and scientists.

The Soviet used stray dogs found around Moscow, partially on the idea that a dog that could survive on the streets there was probably hardy-enough to survive a space program. Dog were relatively easy to care for and train; they were also a good size for fitting into the relatively small rocket payloads. There were tricky specifications, though — only female dogs were initially used, because it was easier to develop sanitation equipment for them. They had to be of specific sizes and even needed to have rather light coats in order to be easily filmed. They had to be trained and developed for the purpose. The caretakers felt strong affinities towards them, as you can imagine, and did not like to see them perish (and many, of course, did).

In later flights, many other animals were sent up as well. In 1960, as part of the flight that put dogs Belka and Strelka into orbit (and brought them back alive again), there were also dozens mice, several rats, insects, plants, seeds, and other biological samples (including strips of human skin). A flight in 1961 also included guinea pigs, reptiles, and human cancer cells, among other critters and specimens. The dogs were, to be sure, the biological centerpieces of such flights, but it's worth noting that, after Laika, dogs captured the international and Soviet public's interest to an unexpected (to the Soviet authorities, anyway) degree. There are a bunch of reasons for this, some specific to Soviet culture (where humans could not be easily commercialized and made into "heroes"), some probably more general (dogs are among the oldest animals to have co-habitated with humans, and we do regard them differently from even other domestic animals).

I have written a bit about this here, with my main source being Olesya Turkina's Soviet Space Dogs, which is a rather popular (but well-researched) book. There are several other sources on the Soviet dog program though, notably Asif Siddiqui's Challenge to Apollo: The Soviet Union and the Space Race, 1945-1974 (2000), starting on page 110 of the PDF here.