Trapping manually is a mindset, mostly lost. For many years trapping has been automated at the plate-making stage for offset printing. For screen-printing, as for flexographic printing, it is best done manually. Illustrator is one of the best tools out there for this purpose.
Ask your printer the trapping amount they want (usually expressed in thousanths of an inch, or in mm). All you really need to do is to "spread" the lighter colors which overlap (mostly to black in your design), into the darker color. In screenprinting, inks are mostly opaque (speak to your printer about this). IF the black, for instance, is opaque, then just continue any red or blue design elements underneath the black, and make black overprint. If the black isn't perfectly opaque, then you might want to only spread the red and blue underneath the black the amount the printer suggests for trapping.
For this design, here's the general texhnique:
Make sure all colors are spot colors.
For the black bicyclists, copy them to an additional layer (underneath the black layer), make them white, and Object>Path>Offset Path the white objects the recommended trap amount (this is a negative number) - release and delete the outer path if it's still there). Then make the black overprint. You could pretty much use this method for all black objects in your design.
For the areas where red meets blue, trapping here will yield a blackish line unless the inks are opaque. Discuss this with your printer. If the inks are opaque, use the same method as above, spreading the blue underneath the red by copying the red object, making it white, and offsetting its path the trapping amount, making red overprint. But if the inks are NOT opaque, it might be better to just not trap those sensitive areas, and let the printer do his best. Sometimes, it's better to use restraint when trapping, as even with bad registration, the mind's eye forgives the printer's slop more than too much trapping.
When you're done trapping, look at your design with View>Overprint Preview, to see if you can live with it.
Thank you so much for your input. I will give it a shot and see what happens. I never thought It would be this complicated!
@DrStrik9 Should you first merge all colors individually before proceeding with the above instructions?
Typically I wouldn't merge colors, as in some situations, things are more complex. It might work in your current design, but in general, when trapping manually, I wouldn't assume colors should be merged to one layer each. For instance, you might have red letters with a blue drop shadow, and blue letters with a red drop shadow, each of which would require a different trapping solution.