There was some interest in ideas for hiking and other outdoor activities in the Victoria area so I've started this thread for people to submit their favourite things to do and places to go! I'll throw some of my favourites below to start:
Hiking Inside City Limits
Mount Doug
There are lots of trails around and up Mount Doug. It is accessible by bike, car, bus, or walking. You can also drive to the top instead of hiking.
Mount Tolmie
A great place to watch the sunset. An easy "hike" (walk) or you can drive to the top. You will see people sledding or skiing here after large snowfalls.
Christmas Hill
Another great place to watch the sunset and get a view of the city. Very easy to get to and takes only 10 minutes to get to the top if you park at the bottom.
Hiking Outside City Limits
Thetis Lake
There are easily walkable trails around Thetis lake. In the summer this is a popular swimming destination.
Goldstream Park
Unfortunately the actual trestle bridge is now closed to the public, but there are still open trails around and you can get a good view of the Finlayson Arm and see some waterfalls in the park. This is also where the Mt Finlayson trailhead is, which is a very fun but difficult hike. In mid-October the salmon will come here to spawn as well.
East Sooke Park
Tons of different trails all over the park, ranging from easy to very difficult. Cabin point is a good target for your first time.
Jocelyn Hill
There are a couple ways to get up to Jocelyn Hill. The most popular route includes Holmes Peak where you can find a swing that is very popular for photos. This overlooks the Finlayson Arm and has a lot of gorgeous views. This a medium difficulty hike.
Englishman River Falls
Near Parksville; 2.5 hours drive from Victoria. Large waterfalls and easy hiking trails.
Sooke Potholes
Lots of trails around a popular summer swimming spot. There are camping areas nearby as well. A must-see for the unique rock formations that make up the swimming pools.
Botanical Beach
A fair ways away from Victoria, but very beautiful. On the ocean near Sooke, there are huge trees, wildlife, and loads of sealife and flora.
Wildlife Etc.
Butterfly Garden
This is between Victoria and Sidney. It is like a large indoor greenhouse where you can walk around and see many different kinds of butterflies and moths, turtles, tortoises, lizards, birds (even a toucan and flamingos!), koi, and more. Most of the animals are "loose" and you can get quite close to them.
Esquimalt Lagoon
This is not in Esquimalt 
This a great place to see seals and lots of different kinds of birds including rarer types of ducks, geese, oystercatchers, and sometimes swans.
North Island Wildlife Recovery Centre
This is a 2 hour drive from Victoria. They have a large collection of animals (bears, many different kinds of birds, etc.) which have been rescued.
Historical
Fort Rodd Hill
Old military base and lighthouse. Lots of history to learn about and very close to the Esquimalt Lagoon.
Centre for the Salish Sea
A small, conservation-focused aquarium and information centre in Sidney. Relatively inexpensive tickets and lots of creatures to see, including an octopus, lots of jellyfish, fish, and crustaceans. Sits right on the coast with picnic space outside.