Renting first is a great idea if you don't know the area well enough to decide where you want to live. Although interest rates will likely rise over the next year, I don't see anything that indicates to me that prices will and knowledge of the area could help avoiding a bad decision as a hedge against any higher costs.
The first question is do you want to be IN the city? If so, there are plenty of 3BR+ places for rent under $4K, but they will mostly be condos or flats with the possibility of a shared yard. If you want a single family your choices will be limited - either need to raise the price or go outside the city.
Outside the city there are a number of areas with downtowns that have the feel of a city neighborhood - Mill Valley, Rockridge, Piedmont, Burlingame, etc... but they are not the same as being IN the city. Either way, most single family homes will cost over a mil for any area you would likely want to be in unless small or fixers.
The best way for me/us to help you is know your budget, desired property type and min beds/baths as this will help direct your search. There are plenty of places in San Francisco where you can get a yard that aren't far from the action, even with a condo. If you are willing to trade a deck for the yard that opens more possibilities.
We work in all the above areas and with both out-of area and international clients, if you send me an email (address below) with more details I can help direct you to what area(s) you will likely be wanting to check out.
And there are companies that handle rentals but craigslist.com will help you get a feel for costs.
Lance King/Owner-Managing Broker