Welcome to the forums. If your battery is good and you've already replaced the solenoid, then your starter is probably bad. Make sure all the cables on the battery, solenoid, and starter are clean and tight.
The next time the starter doesn't turn, tap it with a hammer and see if it turns. If it does, the starter is bad.
Most 4-wheeler problems are caused by a loose nut connecting the handlebars and the seat!!
You only need two tools in life -- WD-40 and duct tape. If it doesn't move and should, use the WD-40. If it shouldn't move and does, use the duct tape.