I'd do some research before trying to put together a petition or a measure. Your city code (laws) should be publicly available; if you don't find it online you'll definitely find it at the city's public library. If you don't have a public library, the city hall should have it somewhere.

i'd look at the last several City Council agendas to see if there are any projects they're working through. Just to get a feel for recent talking points. Also, figure out who your councilmember(s) is(are) and if they have any staff. Shoot them an email with what you're thinking, follow up the next day with their listed phone number and schedule a face-to-face, and then after the meeting follow up with a call at least once a week to reinforce the issue.

If you want to petition, that's much harder since you have to make sure you're hitting voter signatures from the constituency areas of your city council. Are there any good areas where you can expect to get that many signatures? your city code will outline how citizen measures can be petitioned and passed, but it'd be easiest to ask an employee at city hall for some guidelines.

The easiest way to create the necessary legal language is to borrow from the areas that already allow farm animals, which you mention are Seattle and Tacoma. It's likely that the code chapter and paragraph numbers will be similar between Seattle, Tacoma, and the area you guys live in.