day trip from Erfurt ideas

Explore Erfurt tourist website of surrounding area.

Yes, Eisenach to see the Wartburg Castle, and J.S. Bach's hometown.
Got on a wrong train once and ended up in Ilmenau. Nothing there (OK, a regional university), but the train ride was a lovely tour of the Thueringerwald.

There is more than you can see in a day in Weimar, including Buchenwald if you can handle it. Less somber choices include EGAPark botanical garden (dedicated tram stop), the Zeiss Museum in Jena, the Schloss and Schlosstheater in Gotha, and the entire city of Leipzig. We had a car for Eisenach, so you may find local busses to get to the actual attractions to be an annoyance. It's not so trainable from Erfurt, but I would consider changing your itinerary to make sure of seeing Quedlinburg. We went mainly to see the church treasury, but the town deserves several daylight hours. https://en.wikipedia.org/wiki/Theft_of_medieval_art_from_Quedlinburg

You don't mention the rest of your itinerary, but we stopped at the UNESCO WHS of Dessau-Worlitz Gartenreich on our former-East trip.

Only a short tram ride (15 min) from Erfurt is Arnstadt, a beautiful historic town and Johann Sebastian Bach's second place of work. The organ on which he played is still preserved (unfortunately with some alterations).

Some off the beaten path ideas:

Mühlhausen and Bad Langensalza
Both are beautiful small towns and a bit forgotten, since they are in the middle of nowhere. Mühlhausen was actually a Free Imperial City in the Middle Ages and truly important during the Reformation (Thomas Müntzer was executed there). The city offers a big, mostly preserved old town surrounded by a town wall, several nice Gothic churches, and a museum about the Peasant's War.
Bad Langensalza is one of those small, cute Thuringian towns, with a nice little old town, one or two small castles, a few remains of the town wall etc..

Rudolstadt and Schwarzburg (and Saalfeld)
Rudolstadt was the residence of one of those tiny Thuringian duchies and sports a HUUUGE castle high above the rather uninteresting old town. The castle offers one ot two interesting museums, and down in the old town you can find the house where Schiller and Goethe met for the first time.
Schwarzburg was the seat of the counts of Schwarzburg. To get there you follow the most beautiful valley of Thuringia. The castle of Schwarzburg was ruined by both the Nazis and the Commies, but two very interesting parts were saved and restored in the last few years: the armoury and the ancestral hall.
Saalfeld is, again, one of those tiny and cute Thuringian towns, with a nice centre. Highlight of the town are the nearby Fairy Grottoes.

Schmalkalden
Again a small town with a big castle, also historically interesting bc of the links to the Protestant Reformation. The old town full of half-timbered houses is lovely.

Lauscha
Probbly too deep in the mountains to get there without a car, but this is the village where the Christmas baubles were invented. They are still made there. Prince Albert, husband of Queen Victoria, was from this area and introduced the tradition to the English court, who in turn made it popular in the English-speaking world.

Meiningen
Another residence of a tiny Thurigian duchy with a castle, a park and a rather mediocre old town. The real highlight is the Steam Locomotive Works, which has specialised in the maintenance of museum steam locomotives.

Dornburg Castles
Three castles high above the Saale valley., made popular by Goethe, who liked the spot.

Lauscha is a bit of a hassle by public transportation. The town itself isn't especially lovely, but there are Christmas ornaments everywhere and the glass museum is worthwhile if you have an interest in the subject. Note that the inexpensive, colorful marbles sold in the museum shop are from Mexico (at least that's what they had in 2015).

If it's going to be a day trip from Erfurt, then I would get to Weimar prior to noon, spend the day visiting the houses/museums of Herder, Goethe, Schiller, and Wieland.

Another possibility to go Naumburg an der Saale....a real gem like Weimar in German cultural history.

Number 3...see the battlefield museum in Jena, a bit taxing getting out there by public transportation but can be done. it's near Apolda.

Saalfeld/Saale and the Thüringer Meer...
https://www.fahrgastschiffahrt-hohenwarte.de/
https://www.saalfeld-urlaub.de/ausflugsziele.html