In theory it seems doable to hitchhike directly from the centre, as there is the entrance to several motorways. However, the people of the city are not very hitchhiker-friendly, so it is better to directly go to a petrol station on the motorway.
It should be noted that there is barely any traffic around and out of the city on Sunday mornings. This may be the case with other parts of France.
<map lat='48.55388717497782' lng='7.7474212646484375' zoom='11' view='0' float='right' height='400'/> Public transport is very cheap for a western European city. A 24 hour pass costs EUR 4.30, for EUR 6.40 you get a 24 hour pass for 3 persons (August 2015). See fares.
Going east (Germany)
Walking (from Strasbourg centre) to Kehl takes about 50 minutes if you go the shortest route (check Google maps!). Strasbourg tram reaches Kehl center since 2018, the price is the same as within Strasbourg, 2€
Alternatively, you can take the regional train (around 6 minutes for about EUR 3) or the tram (line D, 20 minutes, 1.80€).
By train, for 2 euro more, you can continue after Kehl till Appenweier. From there you can walk or hitchhike to the petrol station "Rasthof Renchtal" (2,5 km check the route on maps.google.com or see Renchtal for details ). It can be a good alternative to Kehl because in 30 minutes more, you're on the German motorway network.
Going North (Metz, Saarbrücken) and South (Mulhouse, Switzerland)
Option 1 Shell and Total fuel stations on A35 highway
Take the tram (red line A) to Parc Malraux and then the bus 13 direction Lingolsheim gare and get out at Ostwald Eglise. From the bus stop walk up to the entrance of the church ("Église" is French for "church") nearby. Facing the entrance just walk down Rue de l'Église parallel left to the church. Keep on walking until you reach Rue D'Illkirch which leads to the bridge over the motorway (see directions). If you want to go south, don't cross the bridge: There is a steep path down to the petrol station. If you want to go north cross the bridge and access the other petrol station.
Heading south you might want to avoid the steep path which leads to the petrol station. Facing the entrance of the church you can also walk to the left, down Rue de Lorraine. At the end of Rue de Lorraine turn right and follow the street until you reach the A35 motorway (see directions). There is a direct entrance to the service station for traffic going south towards Mulhouse and Switzerland.
Toward Mulhouse via the D1083. Take the bus 62 (at the end of the A tram line Graffenstaden) and stop at . There you can walk up north for 3/5 minutes, on the D1083, toward a Total gas station! Eventually the road joins the N83, and the A35.
Going North (Metz, Saarbrücken, Paris)
Take a regional train direction Brumath/Mommenheim and make sure it stops at Stephansfeld. From there you can walk within 15 min to the petrol station Air de Service Brumath on the A4 and start hitchhiking straight at the petrol station. The trainticket from Strasbourg to Stephansfeld costs EUR 3.90 (2016). You may have to jump a fence to enter the actual petrol station, but it is not too tall and in June 2016 there was a convenient hole in it.
Alternatively, just walk (10 minutes from the centre) to Place de Haguenau, and walk to the entrance of the A4 (there is a cycle path that leads right next to it if you don't like contending with the cars). There is a lot of space for people to stop and it is the best way to head north from the centre. Mind you, do NOT hitchhike at the Place de Haguenau as there is absolutely no shoulder to pull over.
Join the A4 by the Going north route see above. Good solution for Paris.
Join the N4 take the "bus interurbain" #205,#206 or #207 from "Ancienne synagogue-Les Halles" or "Gare Centrale" to Ittenheim (eur 1.80). From Ittenheim's bus stop, walk till the entry of the village (3 minutes), and you're on the best spot of the N4 to leave Strasbourg. This road has the advantage to be free of fees so many cool people use it. I recommend it to go to Metz and Nancy cause you don't loose a lot of time compared to motorways and you have many more chances to get a lift.