LYS and Gare de Lyon Part-Dieu are at the ends of the line for RhônExpress, so it's very easy to use. You can save a little money on your ticket by buying it online ahead of time and either printing it out or using the RhônExpress app.
Once at Part-Dieu (which, by the way, is undergoing a multi-year massive expansion to bring it up to a size compatible with current traffic) you have, I suppose, four options:
- Hail a taxi
- Take the Metro, as described above by Tod
- Take a bus (line C9)
No. 1 is cheapest, No. 2 is the most expensive, and Nos. 3 and 4 have a nominal cost (1.90€ for a single ticket, 17.30€ for a carnet of 10, reducing the per-ticket cost by about 9 percent) or you can buy one of the various timed cards, which I've personally never used because I just use my refillable photo TCL card. Information on fees ("tarifs") and routes are on TCL's website. If you'll have a working smartphone, TCL's app is one of the better ones -- much better than Paris's transit app.
My recommendation would be to take the bus -- it's simpler than the Metro, you don't have to go up and down on escalators, elevators, or stairs, and you can see the town as you go from Part-Dieu to Bellecour.
Note that there are several exits out of Part-Dieu, so make sure you take the right one: you'll want the Vivier Merle exit, marked "Porte Rhône." You'll know you're at the right exit if there's an orange La Croissanterie sign off to your right and a blue O'Conway's Pub sign off to your left. If you see an exit labeled "Porte Alpes" with escalators going down off to your left and an Esprit store to your right, you're 180 degrees away from your desired exit.
Anyway, exit out of Part-Dieu and head diagonally across the place toward the left corner. The C9 stop in the direction Bellecour has been moved during construction to in front of the library ("bibliotheque"). Here is TCL's note about the "perturbation" on C9 because of Part-Dieu construction. And here is a photo of the front of the library so you'll know what it looks like. If you need help, feel free to ask one of the TCL employees you'll probably see at Part-Dieu.
When you find the C9 stop, make sure it's the one labeled "Bellecour" because that's (of course) the direction you want to go. If the C9 stop is labeled "Hôpitaux Est" that's going in the opposite direction of what you want. Here's TCL's schematic map of the C9 route.
Once you get to the Bellecour terminus for line C9, the stop is just beyond the southwest corner of Place Bellecour on a street bordering the much smaller Place Antonin Poncet, just in front of the large post office and the distinctive clock tower across the street from post office ("Clocher de la Charité, dating from the late 1600s).
Good luck. Lyon's transit system is good, easy to use, and extensive.