Language classes private

How much do private language classes cost in Stuttgart,say for a1 and a2 levels and what is the procedure for admission?

Google has all of the answers. Not meant as a recommendation per se but I am posting a link to the Agi language school in Stuttgart, which offers private lessons. Lessons are priced in 45 minute blocks, up to 9 blocks for 54 Euros each, 10 to 20 blocks for 49 Euros each or 30+ blocks for 45 Euro each.
https://www.anglo-german.com/sprachkurs … inigruppe/

Here is another website that provides private tutors either live or instruction per Skype (a bit cheaper). Prices vary from (live tutor) 49 Euros per block to 32.50 Euros depending on how many you agree to. Like most such offers, it cost around 10 Euros per person to have an additional person (up to 4 people). So theoretically one could still get pretty personalized instruction but much cheaper if you find another person or persons to share with.
https://www.language-trainers.de/preise.php

For information about admission etc. check the websites.

There are more offers listed at Google or at Ebay small-advertisements or one could even look for a student at University by putting an advert on a pin-board.  One is unlikely to get an experienced professional this way but such tutors should be findable in the 15 to 20 Euro per hour range.