Start Punkt Deutsch is an initiative by the Land Steiermark, Stadt Graz und Österreichischem Integrationsfonds (ÖIF). You can visit their website for a long list of German classes in Graz listed by level, institution and start date. You can also visit their office at Reitschulgasse 19, 8010 Graz for a free personal consultation and get information on which course is right for you (cost, funding options, childcare etc).
Here are some of my personal recommendations:
For small classes and personalised lessons, the love2learn German classes at the Seddwell Center in Gries are very good. They are about to start an intensive german early morning course -Tues, Wed, Thur, A1 8:00 -8:45 & A2 9:00 am – 9:45 – €10 a class plus base membership. They also have on-line German classes on Wednesday morning for B1 & B2/C1.
VHS (Volkshochschule) offers good value German classes for all levels. Especially if you or your spouse works for a company in Styria as you are then eligible for the €60 Bildungsscheck which you can put towards the cost of a class. They also have a great daycare center where you leave your baby / child while taking a class.
Caritas offers intensive German classes with great teachers. B2 and C1 are free for third country nationals!
Urania offers affordable German courses and has good teachers too. Price depends on how many participants enroll, plus Urania membership.
Free Conversation Groups!
The Frauenservice at Lendplatz offers two free German conversations groups for women- Thursday 9.15 to 11.00 and Tuesdays 13.00 to 14.30 (except for school holidays).
Nowa also offers a free German conversation group for women Tuesdays twice a month 9.30 -11.00.
Deutsch um drei offers a free German conversation group for women on Tuesdays at the Stadtbibliothek Nord library branch (Theodor-Körner-Strasse 59, Geidorf). Plus free childcare for kids 0 – 10 years! Registration by Deutsch&Mehr.
I also recommend finding a tandem partner as a great way to practice and improve your German. Tandem Graz Facebook group is a good place to find someone. Or ask around friends, many Austrians are keen to practice their English or other languages.