People typically arrive into Osaka at either Shin-Osaka Station, Osaka Station, Namba Station, Nishikujo Station, Nipponbashi Station or Shinsaibashi Station.
You can use the Hyperdia website or Google Maps to work out train schedule and route options from each of these stations. You should expect to change trains 2-3 times before getting to Universal City Station.
You can use an ICOCA or Suica card to tap on and off each train, and the route is also covered by your Japan Rail Pass.
USJ has a good graphic which shows the specific trains and travelling time required to get from the Osaka / Kyoto region to Universal City Station. All the train stations have clear signage and are easy to navigate with the help of online tools like Google Maps.