By Private Transfer
Duration: 3-4 hours
Getting a private transfer is the quickest and easiest way of getting from Chitwan to Lumbini. The highway passes through Bharatpur, Butwal, and then down through Siddharthanagar to Lumbini, which is close to the border with India. It's not an especially scenic journey, as this part of Nepal is quite flat and heavily populated. It's best to arrange a private transfer through an agent in Kathmandu, as the towns you're likely to be staying in in Chitwan (Sauraha or Barauli) are small.
By Local Bus
Duration: 5-7 hours
There are no direct or tourist buses between Chitwan and Lumbini. The only option is to get a local bus to Butwal (about 4 hours) and then another local bus from Butwal to Lumbini (about 1 hour). Buses from Butwal to Lumbini leave quite regularly, but you might have to wait for a little while.
There is no way to check in advance because local buses in Nepal don't really follow a schedule (at least, not one that's widely known by passengers, or available to tourists!). Local buses are only a good option if you're on a limited budget or really want to experience how the Nepali locals travel—they tend to be slow, over-crowded, and not very safe.