acquire and get are a formal and an informal version of the word for obtaining some specific amount of experience - the amount that makes a difference between "inexperienced" and "experienced". You don't get experience as a shopkeeper over one day of work, you get it after at least a couple weeks.
gain and gather are passive (you do your work and experience comes by itself) and active (you seek out opportunities to gain more experience) of the process.
You gain some experience over a day of work putting plaster on the walls. You get the experience of skilled plasterer over the course of three months. But you will gather the experience of a builder only if you quit plastering and seek out other occupations by changing specializations and getting experience in each of them - learn carpentry, masonry, basics of technical drawing reading, and a bunch of others that comprise the broad definition of a builder.
The borders aren't firmly defined, and people will often "get a little experience" (gain), or "gather experiences of others" (poll), and generally all these can be used more or less interchangeably - what I outlined is just a kind of "hunch" where which fits.