city option on train ticket

No. Here is the Deutsche Bahn website where they show the validity zone for each City-Ticket. If you scroll to Thüringen you will find Erfurt, where it shows that the City-Ticket is only valid in "Stadtgebiet Erfurt (Tarifzone 10)". There is a map there too, where you can see that the zone stops well clear of Weimar. You will therefore have to buy a separate ticket for travel onwards to Weimar.

That ticket to Erfurt ends there. You need another ticket to go from Erfurt to Weimar.

Great that you're going to Weimar, one of my favourite towns in Germany. How many nights are you spending there? Several museums connected with German literary personalities are located there.

2 nights looking fwd

Is there a local rain from Erfurt to Weimar?
Thanks

Love suggestions on places

Deutsche Bahn navigator will give you the time table.

Yes, to find local trains from Erfurt to Weimar, use this link to DB - https://reiseauskunft.bahn.de/bin/query.exe/en - key in Erfurt Hbf to Weimar Hbf; input your date and time of travel (so you get a good idea of the times), then check off the "only local transport" option, and it will show you all the local trains connecting the two.

If you want to see earlier or later times, click the red "up" arrow for earlier, and the red "down" arrow for later.

Other options nearby, Eisenach, Gotha (Friedenstein Palace). Farther, Quedlinburg, but quite a special old town. I wanted to see the Zeiss optics museum in Jena, but that might be the only thing to do there. Didn't have time. If you have the slightest interest in public gardens, go to EGAPark in Erfurt-has its own tram stop.

If you have not been to a Concentration Camp, Buchenwald is a city bus ride from Weimar, but of course a sobering afternoon. Weimar has more to do than Erfurt.

Aside from Erfurt you can easily do a day trip to Weimar should you be considering that, ie, from such towns/cities as Jena, Leipzig, and Naumburg an der Saale, all under one hour distance.