You just got yourself a new Playstation Vita, and are looking for a Custom Firmware solution? Then bookmark this page, as it will be kept up to date with the latest, greatest, and simplest solutions available.
Although there is currently no “native” CFW solution for the Vita, solutions exist inside the PSP emulator of the Vita, which give you various levels of homebrew/backup access. Additional hacks exist from within the PS1 emulator, which lets people play PS1 isos (with perfect sound compatibility, unlike other PS1 hacks within the PSP emulator)
If you can’t run any of those (either because you don’t have the right game, or because you have a recent firmware), an alternative to Custom firmwares is VHBL, assuming you are lucky enough to grab the exploited games when they get announced. VHBL is only a partial solution, as it only allows you to play a subset of PSP homebrews on the Vita.
One drawback is that all these solutions require you to purchase an “exploitable” game on the PSN when it gets announced. We regularly announce names of new exploitable games on this blog, so stay tuned if you didn’t get the games in time. In 2012 alone, 8 such game exploits were released. In 2014, more than 50 were announced.