I guess I am a little confused about wanting a street bike but compromising on a dirt bike. A street bike is transportation, something that has utility. A dirt bike is something you buy for fun.
What you need more than anything else are friends that also have a dirt bike. You really don't want to go out riding by yourself.
As for street legal: There are bikes that are referred to as "dual sport" that are sold as a street legal bike that is also off road capable. The Yamaha XT225 is such a beast, and I know that KTM has some very nice ones as well. You live in California which is perhaps the most restrictive state in the USA as far as street legal registration is concern. Do NOT believe any of the claims of being able to make a dirt bike street legal in California, kits or no kits. It may work in other states but it will NOT work in California.
Not only is California very restrictive for street legal they are also restrictive on dirt bikes. All off road vehicles must be registered in order to ride on any public land. There are two versions of the off road registration: Green Sticker and Red Sticker. Green sticker lets you ride year round. Red sticker restricts your riding, often to just the winter. Since 2003 what has separated Green from Red is the emmission controls that the vehicle has. If you buy a bike that is 2002 or older it will automatically qualify for a green sticker.
While a sticker is legally required to ride just about anywhere in California the enforcement of it is very sparse in the California desert region for the simple reason that there is so much area to cover. If you want to ride in the national forests the enformcenet is much higher. I highly recommend that you buy a bike that has a clean title with the desired "Green sticker". Yo will find a lot of bikes that have been brought in from out of state or for whatever reason have never been properly registered. While it is possible to get the OHV registration it is a hassel, requiring several trips to DMV and at least one to the CHP. Believe me, it is worth a few extra $$$ to get a pink slip!
You will find that there are three types of bikes that you might consider: Dual sport, off road and MX.
As discussed above, dual sport is a street legal bike that has off road capability. To keep it street legal it needs to have street legal tires, which will suck in the desert. The turn signals and mirros suffer a lot when you crash, which happens a lot on a dirt bike. There are a lot of people that enjoy them, however, and there are even competition events that require a dual sport bike to compete.
An "off road" bike are the lower end of the dirt bikes. They will usually have a detuned engine, which makes them easier to start and more reliable. The newer ones will also have the emission controls required for a green sticker in California. They may have headlights, a battery and possibly even electric start. What they lack is the suspension travel that the MX bikes have but if you are not an aggressive rider it may not matter to you.
The MX bikes are all about race. The engines are higher compression and high performance. The suspension will have much more travel. They cost a lot more.
As for the size of the bike: essentially anything 125 cc and above is a full size bike so we are talking engine size only. A 125 two stroke or 250 four stroke has plenty of power to move you around on flat ground and most hills. When you want to climb a really steep hill the smaller engine bikes need a lot more skill to manage.
The sand dunes is one area where bigger is better. Keep in mind that you will want a paddle tire on the back, and that will mean either changing tires or having a spare rim and changing rims.
Since you are just starting out and don't seem to have a firm idea of how you will be using the bike I suggest starting off cheap. Something like a Yamaha TTR-230 is cheap and a lot of fun. Buy one a few years old and you will get it cheap enough that you will have money left over to buy the other gear you need:
Helmet $140
Boots $120
Gloves $ 20
goggles $20
chest protector: $40
knee pads $15
riding pants $50
Tie downs $20
gas can $30
Do you have a truck or trailer to carry your bike around in?
Good luck!
Rod
P.S. Spear fishing, eh? Come north, I'll teach you abalone diving!