It seems that the transmission is just not shifting from second to third because something may be sticking in the internal controls which are accessible by removing the pan. Unfortunately, if you've never been into an automatic transmission before, this isn't something you want to tackle. I would suggest getting references for a reputable transmission shop to take a look at it.
Since it backs up and has first and second, the band and all the clutch packs are functioning so it doesn't sound like you need a complete overhaul but having someone who knows their stuff assess the situation is probably best at this point.
Since it backs up and has first and second, the band and all the clutch packs are functioning so it doesn't sound like you need a complete overhaul but having someone who knows their stuff assess the situation is probably best at this point.