Regular lubrication of your instrument will significantly increase its lifespan, helping maintain its playing condition, value, and lifespan.
Synthetic vs. Hydrocarbon/Petroleum-based
Light vs. Medium vs. Heavy
You should, as a rule of thumb, use the lightest oil on your instrument you can. Excessively heavy lubricants can adversely affect both the feel and even the playing characteristics of the horn. However, an excessively light oil on a worn instrument won't do as good of a job of ensuring a seal, and you may end up replenishing your oil application more often than necessary compared to with a heavier oil. Heavier oils usually do a better job protecting already-worn surfaces from additional wear, and lubrication especially in cases of antique instruments with plating loss or raw brass surfaces on the slides/valves.
Valve Oil vs. Rotor Oil
Generally valve and rotor oils are identical and interchangeable. The main difference is usually in the application tip; usually rotor oil bottles have a dull needle tip, which allows careful application in hard to reach parts of the rotor.
Trumpet Slide Oil vs. Trombone Slide Oil
Never use trumpet (tuning) slide oil on trombone slides, it's much too thick for that application.
Trombone Slide Cream (e.g. Trombotine, Superslick) vs. Slide Lubricant (e.g. Yamaha, Slide-o-mix)
Slide cream is intended to be used with regular spraying of water on the slide to "activate" it (i.e. get it to a good working consistency) and can dry out between uses. It is essential that you use a spray bottle in combination with these lubricants or the feel will be way too heavy/sticky.
Slide lubricants, such as the one-part Yamaha or Slide-o-Mix, or the two-part Slide-o-Mix are intended to be used without needing a water spray bottle on hand. They will stay consistent in viscosity between playing sessions. However, a spray bottle can always be used to adjust the consistency if needed.