Gabe,
Welcome to the site. Glad you are considering a Rocket. They are an intoxicating bike to ride and own.
As to problems, it seems all makes of bikes have problems that are systemic to that model. The Rocket is no exception. The older models (pre 2009/10) had electrical issues, problems with the torsional damper bearings, cam chain tensioner premature wear, clutch lifter shaft and 2nd gear issues that were resolved (for the most part) with an upgrade kit Triumph supplied, some under warranty, some for a special price. A lot of guys took advantage of the offer from Triumph but there are a pile of pre 2009 bikes still running that were never upgraded with no issues. I have a chum that owns (he claims) one of the first Rockets into Canada. The only issue he had was a defective Throttle Position sensor.
For the later models it seems there were some Shift Detent Spring issues causing units to jump out of gear or not stay in gear (this was a problem on all years). Also, a number of Rockets appeared to have weak snap rings in the tranny which could cause the gear train to move on the shaft sometimes engaging 2 gear sets at once. A lot of guys had this tranny problem but many many more had none, me being included. You usually hear from the guys that have had problems and the tranny issues seems to stand out, but there are a lot of owners that haven't had any issues with thousands of miles on their rides. If you are looking for a Rocket, I'd still pick a later model unit since they have had all the latest upgrades. You rarely hear of electrical, cam chain or torsional damper issues with these bikes. A lot of the issues we do hear about are from guys solving problems modding their bikes.
Now if you are handy with a wrench and can get an early model with an upgrade kit to play with... go for it. They are dirt cheap. Looking for new..... These guys have at least 6 left for $10,999
National Powersports Distributors