National parks are a bit far but you have many options depending on how long you are willing to travel. Here are a couple options closer or further from Lyon.
Parc National des Écrins
This is far from Lyon and better done from Grenoble (1h30 from Lyon), and it is the beginning of "real" mountains, the Alps. This is one destination option among many. You can reach Bourg d'Oisans, a gate to the Ecrins national park by first reaching Grenoble by train with SNCF (1h30). Then you can take the bus 3000 towards Bourg d'Oisans for another 1h30. The earliest you can get there seems around 10.30 AM on weekdays. This is considered late in summer (most hikers prefer starting early, around 7-8 AM). From there you can hike in the Ecrins national park, for example start a part of the GR54 (GR is for Grande Randonnée, i.e. long hike). There is a map of the whole park to find ideas of other places. Grenoble really is a gate to the Alps, but the issue is the length of the trip from Lyon. The last bus from Bourg d'Oisans is at 7PM and you can be back to Lyon by 11.30PM. This gives you 7-8 hours to actually hike, if the sun goes down late - but this also means more heat.
I encourage you to explore the bus lines of Transisere from Grenoble to find other accesses to trailheads. There are many resources to find trails and national parks, like Geoportail.
Monts du Lyonnais
That's my hometown area. These are hills located at the southwest of Lyon, and you can get an elevation difference of several hundreds of meters when hiking or biking there. There are many options again. You can find an example of itinerary on the Rhone department website for mountain bikes but it's possible to hike it. There are GPX traces from a local mountain biking club on openrunner (search for user Bikets_Mornantais). To access Mornant and the area, you can take a coach of the Rhone department, the bus line 14. Bus tickets cost 2 euros and can be bought on board. There are few buses so plan ahead (and bus times are not so reliable if you don't board at the terminus).
If you do not want to take too many risks with French transportation systems (and I would not blame you), you can stick to TCL, the Lyon urban buses. Take the line 40 from Bellecour towards Neuville, stop at Rochetaillée, cross the Saône river to go to Couzon-au-Mont-d'Or and climb to the Mont Thou. There is an itinerary on openrunner starting there. It is also accessible with a regional train from Part-Dieu to Couzon-au-Mont-d'Or.
Overall, you have many options close in the hills or far in the Alps for hiking/biking day trips around Lyon. These are only examples and I really encourage you to check openrunner or other hiking itinerary websites to find some, as well as geoportail for maps or the Rhone department website. You have multiple ways to go around: TCL for urban buses, Rhone department buses or regional trains. Trailheads are not always easily accessible but with some patience you should find something you like.