Character choice is a very personal thing - in VERY general terms, Tactical focuses on dps, Engineer on healing and Science on buffs and debuffs. In the end it's only part of the puzzle, your play style and ship and bridge officer powers can work with any captain type. Personally, I prefer Tac and Sci (the latter is very versatile and can be a lot of fun).
Ships: don't worry about it until you hit level 50/60. All the lower levels go by so quickly that you will not keep the ship for that level very long. Use them to try out different types, so that you know what you want by the time you hit end game. Although the game goes up to 60 now, I'd still consider 50 the point-of-choice, as the last 10 levels take a lot longer to complete and a lot of the end game stuff unlocks at 50.
Other tips:
- Play through the new tutorial; it's actually interesting these days.
- Don't forget about the duty officer system once you unlock it. It can be a great source of income while you're offline.
- Find a fleet that's open to helping new(ish) players. Pick one that caters to your time zone if possible.
- There is no such thing as "too much dilithium".
- Have fun.
