How do I get my atv to accelerate?

Have you checked the Crankcase seals? I have 2 Big Bosses with 250's. 1989 + 1992

Both were terrible at starting and idling....after much troubleshooting I finally changed the crankshaft seals. (Which btw looked fine from the outside!) machines can be Cold now and will start 2 or 3rd pull with the cord now. Idle perfectly.

Also the oil injection system stopped working intermittently. Machine would just quit.....crank and crank it forever. So I just started using 32/1 mix and unhooked the Oil Injection cable from the Throttle.

sounds like fuel flow from the tank is having trouble and its running off the gas built up in the bowl. could be the petcock, tank strainer, tank vent clogged

did you adjust the valves? check and make sure the fuel pump is putting out enough fuel, way past time to replace it. check your voltage too, the regulators in those are famous for dieing.

backfiring is usually a sign of an air leak in the intake manifold

these ATVs do a lot of hard work for size of motor

worth checking other things as per normal servicing
new spark plug
clean air filter
drain and add new fuel to tank
wont hurt to clean the carby or bowl contents
if fitted with a fuel filter inline .. fit a new one

and valves adjustment if still back firing

Have you checked the air filter. it may be clogged and requires either cleaning or replacing. if it is a sponge filter, try washing it out with petrol, and put back in when dry.
if its a paper filter, it should be replaced if dirty.
hope this helps, i am sure it will cure the problem for you.

Spray carb cleaner staight into carb opening at fast idle

#1 completely clean and set the carb..make sure air filter is clean.

Check spark plug and air filter and fuel filter.If not, you may need carb. cleaned. Also check carb bowl vent hose for restrictions.