A really really grassroots way for campaigning against animal testing is creating awareness! The best way to halt animal testing is to stop its funding, so make people refrain from purchasing products which do test on animals. This can be done in heaps and heaps of ways. Hand out fliers on a busy city corner? Hold a stall at a festival and collect signatures for the petition? Drop leaflets in peoples letterboxes?
The best way to really drive the idea into someones head is if you contact them personally, so they have a feeling of guilt whenever they buy their procter & gamble, l'oreal, etc. If you're really serious, it would be incredible to hold a stall
You could also send lots and lots of letters and emails to the headquarters of companies that DO test on animals, informing them that you will refrain from purchasing any of their products and will advise others to do so as well.
And support companies that DONT test on animals. Write letters to them saying that they're cool

I sent christmas cards last year (lol yes lame i know, but funny)