EVE New Citizens Q&A

So I just converted to Omega for this month at least. However I have some questions about what goals to go for. I'm currently doing factional warfare and such and I wonder if tactical destroyers are a good idea? I'm thinking about training the skills but it will take a long time, so I have to know if it is a worthwhile goal or not.

T3ds are pretty great. They took a bit of a beating a couple weeks ago, so now they rely more heavily on switching roles mid-fight.
There are a lot of roles in all fleet/gang sizes filled by ships in the dessie-down line. SP here is not wasted.
Plus, t3ds help take some of the pain out of training into interdictors.

T3ds are good. Go for it.

Keep in mind that the more advanced your ship, the higher your support and weapon skills have to be, otherwise your experience will consist of just losing more expensive ships. It takes 3 squares in the support skills to unlock T2 defenses, so that's pretty fast, but T2 weapons can take weeks / month. And weapon quality tends to matter in PVP.

There are multiple things you have to "grow" at the same time: you can unlock more ships, but also you need to keep up with the support and weapon skills for them, and also your ability to make ISK and afford to lose these more expensive ships, and finally in certain cases like FW or agent missions, your rank/standings with the NPC's.

EDIT: Brief explanation of the various ship classes:

T1 ships have "versatility" as their gimmick. You can fit them to pack a punch, you can fit them to be sneaky, fast, tough, whatever. These ships are excellent for pick-up fleets or if you don't know what the enemy is bringing.

T2 ships are super-specialized. They take one thing to the limit (speed, cloaking, jamming, repair, whatever), and pay for it with huge vulnerabilities. T2 ships require an organized fleet, where everybody has a role (DPS, defense, repair, jamming, etc.) and the fleet members are really good at coordinating. You need a good fleet commander (FC) to handle the tactics well.

T3 ships are gimmick-ships; they have one trick or ability that comes with the ship, such as an immunity or various modes of operation. For these ships you need to either: know the game mechanics very well, or have a fleet pre-condition the battle scene so you can apply your ship's gimmick.

Capital ships are NOT solo ships; they are extremely expensive and thus demand a lot of skills to be trained to 5 (you don't want to lose your carrier because you didn't train that last 5% of shields, for example). Typically used in 0.0 for large scale wars.

Frigates rely on speed for defense. Destroyers have frigate defenses but bigger size so they're easy to hit with bigger weapons. Cruisers rely on actual armor or shield tank, but their thing is super-versatility; they don't have the 6+ slots required for a solid tank. Battlecruisers are "attack" cruisers, focusing on doing huge damage before they can be destroyed. Battleships are bricks, not mobile (that's their vulnerability), but they can put out a huge amount of damage at ranges approaching 150-200km and have a solid tank for defense.

Interesting points, is loyalty grind at plexes enough to make the isk necessary to maintain something like this?

Also, for a Hecate for example, what weapons and support stuff should I start training? Hybrid turrets and armor?

All the ships are good, t3 dessie among them. What you need to decide is if the playstyle of the ship fits your playstyle. You may feel like you'd have a more fun time in a frigate or pirate cruiser.