It's only a 3-hour drive, round trip, but there's a lot of beautiful places to see between Avignon and Apt.
Head east out of Avignon on the N100 (direction L Isle-sur-la-Sorgue). You should pass through the villages of Chateauneuf-de-Gadagne and Le Thor. It's a bit tricky finding your way out of town the first time, but if you get it wrong, you don't need to go back: just turn left (north) or right (south) to l'Isle-sur-la-Sorgue.
l'Isle-sur-la-Sorgue was once named "L'Isle en Venise", attesting to it's reputation as the "Venice of Provence". The town is laced by a maze of canals and several branches of the Sorgue river. Several large water wheels are quite picturesque, and you'll find markets in the old town. Antiques: Le Quai de la Gare is a group of 30 antique shops in a single location, open on weekends and holidays, and there's an antique market on Sundays in the old town.
Go northeast from l'Isle-sur-la-Sorgue, and immediately right onto the D25 (direction Fontaine-de-Vaucluse). It's a very short drive, part of it alongside the lovely Sorgue river. You'll have to pay to park, but that's sometimes the price of popularity (theirs, not yours).
Fontaine-de-Vaucluse. The key site here is the full-fledged Sorgue river that gushes up from the ground into a deep pool surrounded by rocky cliffs. Various attractions include a 15th-century paper mill, ancient aqueducts and old buildings, and possibly too-many souvenir shops.
Alternate. Saumane-de-Vaucluse is only 4 km away, easy to get to either on your way to Fontaine-de-Vaucluse or when you're leaving. Leaving Fontaine-de-Vaucluse to the west (back the way you arrived), after passing under the aqueduct, turn right on the little D57 road to Saumane-de-Vaucluse. The village is lovely, small and isolated but with lovely walks with a lot of green and a lot of flowers, and a private chateau where the Marquis de Sade once stayed.
From Fontaine-de-Vaucluse or Saumane-de-Vaucluse, loop south, east and north to Gordes; on 15 km away.
Gordes is perched on the edge of a cliff top, and it's worth coming here just for that view of the village. A unique site in Gordes is the Village of Bories (entrance about 5 euros, and definitely worth it). Several museums, art galleries, shops, resturants and cafés, and nearby lavender fields are some of the things that draw the visitors to Gordes.
Roussillon is only 8 km east of Gordes, and there are a variety of small roads between the two towns. but
Roussillon is named from the red-ocre color (Latin Vicus russuls) of the land it's built on, and built from. More touristy than we like, but it's a beautiful place and a "must" to see if you're in this part of Provence. Ocre is the color of Provence, and not always red; ocre can range from pale yellow to a rather darkish red, and just about everything in between, and is the basis for that same range of colors in the houses of Provence.
East and southeast from Roussillon, you join the N100 highway and continue on east into Apt, only 11 km total.
Apt was a Gallo-Roman market town a couple of thousand years ago, and it's an interesting market town today. Here you'll find ancient monuments, old walls, good shopping, restaurants, and streets to explore. Apt is billed as the capital of candied fruits (fruit-confits).
Alternate. St Saturnin-les-Apt, 9 km north of Apt, was the home of "the father of truffle raising", has one nicely-restored old windmill, and is a very interesting little medieval village.
Alternate. Saignon is only 4 km southeast, perched on a hilltop overlooking Apt. More restored and more frequented than St Saturnin-les-Apt, with a great view and nearby bories and lavender fields.
Return to Avignon.
You can return directly to Avignon (53 km, 1h00): east on the N100 for 24 km (2 km past Coustellet), then bear left onto the D22 (still the main road, and clearly marked for Avignon).
Slower but much more interesting is to go southwest from Apt and return via the wonderful little Luberon villages of Bonnieux, Lacoste, Ménerbes, Robion, and Cavaillon.