We have been adopted by a new cat in the neighborhood. I'm sure she is here to stay. I's like to know what the most effective and cost friendly and technique I can start with

Technique to do what? Are you wanting to keep her? Technique to feed her? Tame her? Potty train her?

If she's to stay outside, you will need to stick to a monthly flea regime. Speak to your vet about advantage or revolution.

Yes, if she's going to stay outside, the monthly treatment are your best idea. If u want to bring her in the house, you still want to use the monthly treatment at least until fleas are gone. Once you bring her in, its also a good idea to "bug bomb" your house and vaccuum thoroughly after. Empty your vaccuum cleaner outside immediately after that too.