Click on the button in the lower right hand corner for manipulating housing items. Then click on the pet. If your pet flies, for instance a dragon, click on the pet's shadow on the floor, not the pet. At this point I can't remember, but I think you have the option of either moving your pet or picking it up.
Some times you need to move a pet if you are trying to put down a furniture item and the pet is in the way.
Then get out of the housing manipulation by clicking on the button again in the lower right hand corner.
Then access your back pack, the easiest is by pressing "B". Then click on the pets tab, click on the pet you want, then click on the fist to "equip".
If you also have a mount, you can't see the pet outside, if you want to show it off, you will have to get off your mount/broom.
I wish KI would make a "show off" mount, you move +40% but you have your pet showing, and you could swim with your pet, just 40% faster.
