Hi Ruberio,
Ok so Shellac and Gelish are both uv cured manicures done on natural nails (you can get them both over nail enhancements as well)
Gelish is a gel polish and Shellac is a power polish. I don't use Gelish in my business so I can only tell you about Shellac!
Personally speaking I feel Shellac is an amazing product and the best thing about it is that there is absolutely no damage to the natural nail, there is no filing of the nail plate either to Pre application or to remove and it removes quickly and easily. It goes on thin, so doesn't look like a thick overlay and has amazing shine that lasts perfectly. I wear it constantly and always have my nails fairly short. I get weeks out of mine but obviously your natural nail grows so regrowth will be visible after a couple of weeks at which stage you'll want to get it redone.
Some brands point out that their range lasts 3 or 4 weeks, I've had people last 6 weeks with Shellac and not one chip in site. You have to consider how fast your nails grow so with any long wear manicure once the regrowth is up it will be time to change, so your wear time will be the same irrespective of which one you go for.
Neither will add strength to your natural nail but will provide a layer of protection. I'd highly recommend CND Solar Oil and use twice daily to help deeply condition your nails and you'll find it dramatically improves nail health.
Shellac is in my opinion the best out there so that's what I'd recommend!
Let us know how you get on and enjoy your first manicure!