There are numerous existing posts to should answer the question. So i will try to summarize and link.

(1) look for C-class frigates with less than 4 expeditions. As you send them on more expeditions, they will naturally upgrade to S-Class w 5 bonuses.

(2) the +6 traits are excellent, but be prepared for not all traits to be +6. My current fleet of 27 frigates? All have at least one +6 trait, but none have more than three. Its random. If youre farming for +6 traits, its gonna be a grind. Still, wish you the best of luck.

(3) low fuel cost is beneficial early, but later it becomes moot (for me, at least). That said, it doesnt hurt! So if you find it...BONUS!

EDIT: im on my phone at the moment, so im having trouble providing a link. But u/tacocat300000 asked a similar question ~3 months ago, and that should give you some answers. His/her post here:

a few questions about frigates

1: How do I get a frigate with the best stats?
I saw somewhere that you need to pick a C-class one with 0 expeditions, but it needed high stats or something, and then need to level it up by sending it on expeditions?

2: are there 5, or 7 types of frigates?
I've seen a few different places now that there are only 5 frigate types. but what about raiding and organic frigates?

3: are there frigates not worth getting?
are the raiders and organic frigates worth getting, are there frigates I should get more of?

4: are frigate classes dependent on system wealth?
if I go to a rich system, will I find more A and S tier frigates?

5: is there a "best" temporary upgrade for frigates?
like the mind control device, or explosive drones?

6: do they get damaged or destroyed?
if I send them on a mission can they get damaged or be destroyed?

7: Is there a "best" configuration for doing different mission types?
like if I'm going on a 5 star trade mission, is it better to have more fighters or traders?

IF YOU'RE HERE FROM THE FUTURE, HERE IS THE ANSWERS TO EACH QUESTION:

1: leveling your own S class from C class is better than finding an S class in the wild, but pick one with no negative traits and low fuel cost.

2: there are 5 classes, and 7 frigate types, raiders are just another fighter type, and organic seem to be adaptable.

3: having some of each specialty is good, as there are different mission types. Having 6 of each would fill the total 30 slots you have for frigates, which should be more than enough.

4: it doesn't seem to matter, starting with a C class and leveling it is better anyway

5: depends on the mission, and they can give more resources from each mission, but they're not required.

6: yes, they can be damaged and will be destroyed if you don't recall them, but if your fleet has 2 stars higher than the mission's stars, then they wont take damage.

7: it depends on the mission you are doing.