In most cases you will likely use "after". You should only use "right after" when you want to emphasize that the action was immediate. "right after" means the action was only a moment later. "after" simply means that the action came later but we don't know how much later (maybe 10 minutes, maybe 10 hours, maybe 10 days). I went to bed right after I ate dinner. I went to bed after I ate dinner.

right after i finish my dinner i will tell you the correct way to you use right and after :P

right after = just after "I went home right after he called" = "I went home immediately after he called"

right after, means immediately afterwords, after is just in general.