This sounds like a great project, and indeed multi-stage lifts have been useful to many previous FRC games. 2018 was a particularly good year for this as well as 2019.
There are different ways to provide the first stage and following stage motion. If you also want to lift, a common option is to drive the first stage directly with chain and then couple following stages with chain. This can be seen in 1323 Madtown’s 2018 bot. That said, there are also plenty of lifters that are cascade (continuous cord through stages) and have a single robot lifter on top like 1678 and 118 in 2018.
The most common lift cords are probably 3-6mm AmSteel or Dyneema cord, #35 chain, and non-continuous 5mm HTD belting. Our team learned the hard way that 5mm HTD belt does not react well (breaks) to shock loads, so this can be detrimental to building lifts if enough car is not taken in other mechanics or programming to prevent the shocks. The cords are popular because they are easy to work with (esp compared to steel cable), light per weight capacity, and low stretch under load.
To get started, both 1678 (in 2018 and 2019) and 973 (Greyt Elevator) have posted CAD models of their previous bots with very well performing lifts. Both regularly use 2x1" aluminum structure.
1678 in 2018 w/ buddy climb
1323 in 2018 w/ buddy climb
333 in 2018 w/ 2x buddy climb
118 in 2018 w/ belt
5026 in 2018 w/ belt … just for fun because this isn’t really 2 stages and used a steel cable to climb