As a rule, buying a bribe is a sledgehammer approach to reputation adjustment and it results in pretty dramatic changes to one's rep sheet. Of course, it is extremely fast and does not require a ship capable of fighting, so traders can do worse than buying one good bribe that gets them around.
Flying missions for the faction whose favours you desire is also
very effective and it has the advantage that you can stop once the desired effect has been achieved. Otherwise it is similar to the bribe method since other factions - allies/enemies of the faction in question - are affected in exactly the same way as with a bribe.
The third option takes the longest but gives you the best opportunities for fine-tuning your rep-sheet: to improve your reputation indirectly by affecting allies (via bribes/missions) and/or enemies of the faction (via object destruction). This gives you much better choices for shaping your reputation overall since most factions have several allies and enemies. Destroying depots is much more effective than destroying ships but not all factions have depots; also, depots respawn only at server restart and certain depots that are in high demand (police, Bounty Hunters) tend to be kaputt all the time since everybody wants to tune their rep sheet now and then.
That leaves destroying enemy ships. It does not matter if you do it on your own time or during a mission, but if the mission is for an enemy of the faction you want to get green with (e.g., Bounty Hunters in the case of the Corsairs) then the fact that you work for another enemy will reduce the positive effect and it can even cancel or reverse it. In that case it is better to take the missions from a faction that is neutral (e.g. Rheinland house law in the case of the Corsairs) or to do the work pro bono. If you choose your 'workplace' well for the latter then it can be much more lucrative than flying missions, because of the income from looted weapons, ammo and commodities.
The Red Hessians are universally liked for rep adjustment purposes because they are such utter pushovers. The same is true for the Mollys to a certain degree but while it is fun to battle Hessian Stilettos/Sabres in a Crusader or Barracuda the Mollys are too weak even for class 4 fighters like Dagger or Defender. Nice for freighters, though.