No matter what you would skill up, it probably won't help you. Exploring null sec is not only a matter of skills, it is a matter of knowing the game, information and also being lucky.
The hardest one is the first 2-3 gates after high/lowsec. That is where all the gankcamps are and where the most kills are done. After that, you will mostly encounter empty space where you may meet some people, but unless they are in a gang, they won't attack or hunt you. Exceptions do apply, but my experience is, that most rather run than fight, especially if there are 2 of you. The only real danger is landing in a gatecamp. And that is most likely bubbled and there are many people, so you cannot outgun them anyway, so rather go for speed.
More information about how to build a nano ship:
My Tipp would be to use destroyers or Frigates of any race and mod them for speed with a mix of Nanofiber Internal Structure and Overdrives and put some Afterburners on it, or a MWD. Afterburners will still work if you are scrambled, which hinders you from warping or using a MWD. It is up to you to have defense against so called "bubbles" or against single ships. With an Afterburner you might burn out of scrambler range and still get away. With a MWD you might be able to escape a bubble. But it is highly unlikely, that you do if there are actually people around since every PVP fit has a scrambler included.
The only thing that would actually be really useful, would be a T2 Stealth Ship, but Trials cannot fly those.
Besides all that, I strongly recommend reading this article:
It nails it down to the following points:
- Make Safespots
- Be aware where you go and scout the area if possible (unlikely in your case)
- Use your directional scanner and be careful where you go
- Know about the alliances occupying the space and their rules.
The most important point is rather simple though
- Be prepared to die. A lot. And I mean that.
If you managed to get past the initial Gatecamps and are actually in Nullsec, you won't be able to do much in your frigates, I am afraid. Pirates are flying everything from Frigates to Cruisers to Battleships. But nothing you would be able to train in those 21 days would make a real difference.
Seasoned Players travel to Nullsec in a stealth ship or cheap frigate and then have their combat equipment shipped down in a carrier. And after that initial travel, everyone is using Jumpclones. So nobody really travels a lot.
This is why it is relatively safe down there, apart from the really inhabited systems. I was able to live in a system for 3 months out of my T3, before Red Alliance actually discovered me and hunted me down.