When you say you can't get past deserts, do you mean that the sandstorm prevents you getting to the other side?
If so, you need to hang around near the edge of the desert and wait for the sandstorm to pass, you can then pass as soon as you notice it stop. From experience on the PS4, I know that can take a while, but they do stop eventually.
Also, sometimes the wind changes direction during the storm, and you can use that opportunity to make a dash across. A decent weapon is recommended though.
Another option is to build a platform bridge say 50 blocks above the desert, which will allow you to avoid the worst of the enemies. Just start building it well outside the desert, and preferably high enough that the wind doesn't affect you.
Not all worlds have open caves near spawn, though having made 50 worlds, at least one of them should have. To be honest though, I prefer worlds without open caves near spawn, as that's where I have my main arena and caves/holes cause enemies to get stuck there.
If there are no open caves nearby, you can just choose a spot for your hellavator and start digging it. Once you've gotten a little way down the caves should open up a bit.
1.4 should (hopefully) be coming in the next few months. I have noticed from playing 1.4 on mobile is that sandstorm happen far less often.