Annecy to Lyon to Paris

Not sure why you would be worried about connections from Annecy to Lyon unless
you haven't looked into it....

Flixbus, amongst others, runs direct buses, and the SNCF trains run direct as well.

It's ~90-120 minutes one-way.

Lyon certainly has things to see, but you would especially want to stay for dinner, which
does not seem to be part of your scheduling. Also, most of the stuff to see is on the west
side of the Rhone, and Part-Dieu is on the other side; definitely a taxi or subway ride to
reach it.

I think the tradeoff is how much time do you have in Paris; leaving Annecy in the morning
should get you to Paris early afternoon, and you have a half day in Paris. If you spend it
in Lyon, you may have some additional overhead getting around (Flixbus goes to Perrache
and the trains to Paris leave from Part-Dieu, for example), and then there is the hassle of
storing your luggage for the day at the station, and you get to Paris late in the evening
and then need to get to your accomodation from Gare de Lyon.

I used the BlaBla Bus from Annecy to Lyon last June. It was about an hour and a half, the bus was clean and pleasant and Flixbus was also a choice. The train station is next to the bus station in Annecy but from memory, when I was looking for that route the train took longer? Regardless even with an early bus you'll lose a chunk of your morning getting there
As said, the Lyon stations aren't really where you'd likely sightsee, so you'd need to store your bags? And use taxis to save time - at least the two times I did they were way to expensive! Maybe you are considering an early evening train on to Paris?
If you know what you want to specifically see in Lyon that also is a factor? Lastly, not sure your itinerary but maybe you can take a day from Annecy for an overnight in Lyon?
What beautiful cities! Safe travels!

I'd save Lyon for another trip where you can give it more time. We've visited Lyon on three separate trips, 2-3 nights each time, and we always leave wishing we had more time there.
What do you hope to accomplish in half a day? Maybe if we understand why you want to visit, we can tell you if it's feasible.

Thank you everyone. I will save Lyon for another opportunity when we have more than a day trip.

Sorry we poured so much cold water on your idea, but.....

If you went to Lyon early in the morning, spent the night there, and then
took the train to Paris early the next morning, you'd have the better part of
a day in Lyon, as well as the evening.

Take a train to Part-Dieu, drop your bags at a hotel by the train station,
and make dinner reservations ahead of time at a bouchon.

You'll get to Paris before lunchtime, which is still a tradeoff, but gives you
a better shot at seeing Lyon. Probably also saves a little $$ on hotel (Lyon
vs Paris) for that night.