Greater Victoria is actually a collection of municipalities, with Victoria being the major downtown core.
While downtown Victoria may have rents at the rates quoted by Jackie, many of the other municipalities have better prices.
**Keep in mind that there is a major construction project happening here, starting in late 2016 and running to late 2018 that will grind traffic to a halt at the TransCanada Highway and McKenzie Ave intersection because many people work in the city but live in Westshore (a group of municipalities including Langford, Colwood, and I think View Royal, Metchosin and possibly Sooke are under that Westshore umbrella as well, but I'm not sure). There is no train or subway. There are buses, but they can be awkward to connect with depending on where you live in Westshore (I fully expect BC Transit who operates our bus network to bolster service to the Westshore communities for the duration of the construction). In the meantime, if you don't want to waste 3hrs per day commuting 15kms each way, then you won't want to live in Westshore.
IF you need to be in the downtown core often, or if you are planning on going to UVic or Camosun College (Lansdowne Campus, or the Interurban Campus), you would be better to find a suitable place to live downtown (expensive) or in Saanich (moderate), Oak Bay (possibly expensive), Fernwood (moderate), James Bay (possibly expensive), Fairfield (moderate), Uplands (really expensive) Gordon Head (moderate-expensive), Esquimalt (moderate, but I would not recommend the area). With those examples, I'm assuming that a one bedroom place at $1200+ is expensive. Gordon Head is close to UVic and Camosun Lansdowne, so landlords in the area are more familiar with dealing with students.
If you DON'T have to go through that intersection that is being rebuilt over the next two years, then living in Westshore is actually pretty good because it's a newer/more progressive area and most houses have suites built into them. You can rent a 1 bedroom for $1000 or less (and as the traffic becomes a nightmare, those prices may actually drop a bit because living in the area and working downtown will become less feasible for people). That said, as mentioned by the others, vacancy rates in the city are REALLY low.
Now, as for pets... people here seem to be more open to people having a cat or two. More than that becomes a problem because no landlord wants to rent their residential unit - that is a significant expense for them to own (a 2 bdr condo starts at at around $250k) - to someone who wants to live in an animal shelter.
Cats are generally preferable to dogs because there is less noise, they're generally smaller, and cats bury their poops (so a home owner won't be stepping in dog poops by a neglectful tenant). Also, being on an Island, many homes have limited property sizes, so there isn't much space for dogs to do what dogs do (run/jump/play), unless you live further out of town. That isn't to say that people won't rent to dog owners, just that the dogs tend to be smaller and the available places tend to be limited in a market that is already limited.
In addition, landlords may ask for 'pet insurance', meaning that if your cat claws up their carpet, your insurance would cover replacing that carpet when you move. If memory serves, this runs about $250/year... but it's been a while since I looked into that. This would be above your damage deposit, which most landlords are sure to require.
Outside of all that, this isn't a cheap city to live in. It's not as bad as Vancouver, but Victoria is a tourist town and a lot of the downtown and surrounding areas have prices that reflect that.
Hope that helps!
Source: I'm a long time resident of the city. We have a 1-bdr suite in our house with internet, cable, heat, and hot water provided. We have shared laundry machines. Our suite is currently rented for $750, and is under market value by $100-150. Our tenant is a single mid-twenties woman with a cat.