1

Everyone who drives the car must be authorized to do so by the rental car agency. They may charge a fee for this. Make sure and ask and include it in your estimation of charges. Any accident with an unauthorized driver will be hugely expensive.

2

Have you looked at a map? You could drive from San Diego to SF in a day. A boring drive on I-5 but doable. So let's get down to basics. What so you want to see? Deserts? Mountains? The mighty Pacific? Giant Redwoods?Urban sprawl and smog? Disneyland? Zoos? Museums? Small Towns? Farm land? WHAT? Countless car rental companies each with rules about 25 year old drivers. Google American car rental companies. You WILL need to pay for insurance so include that in your budget NOT too mention gas costs and at this very moment we, in the late great state of California, have the highest gas prices in the nation. Over $5 in some places. So, repost what you want to see of California in 13 days. Big state lots to see.
Give us a clue.

3

You are going to have to search around and check a number of rental companies, to see where you can get the best rate.

Check around before you rent at the San Francisco airport. Airport surcharges can add significantly to the rental costs. However, you may find it easier to rent without a one-way drop off fee if you rent at the airport and drop the car off at the San Diego airport.

I assume you will be in SF for a few days at the beginning of your trip. You do not need a car in SF itself, so that will save you some money.

Another important question--what time of year? Where to go and what to do can very much depend on when you will make the drive.

4

With that much time, I'd loop North and take in Wine Country for a day or two, plus some of the Northern Redwoods. You can get insanely good car rental deals with local companies outside of major cities. I've rented a car for $20 a day in Santa Rosa, for example.

13 days gives you so much time. You can basically see most of California. I would try to really spend some decent time in SF, drive Route 1, stop in Santa Cruz, spend time in Big Sur, see Hurst Castle, consider a good hike somewhere, and maybe if you get tired of the big cities, check out Joshua Tree Desert or other parts of the Mojave or Death Valley (I'm more interested in California's nature than our cities, but that's just me, perhaps).

5

If you don't already have auto insurance that will cover your US rental, you may be able to get better rates by using a foreign rental car broker, rather than going directly to the rental agencies. Compare for yourself.

The two scenic routes between southern and northern California are along the coast via Hwy. 1 and along the eastern Sierra via Hwy. 395. If you are doing a round-trip, you can do both. If you are going one way it will depend on your interests (ocean vs mountain/desert), though most people prefer Hwy. 1. The fast route is I-5 through the Central Valley, as #2 said, but it's boring and not scenic. Since you are not in a rush, avoid it.