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You're lucky because IT is not a profession where a big knowledge of the local language is so required, but probably your degree will have to be checked by someone to say if you can work on that profession, Germans take seriously documents that prove that you are allowed work.
You will meet people normally anywhere, work, gym, cafe, etc. Germans are not hard to befriend but they are not going to be bff  in a first moment, actually, don't expect them to be very attached.
There are gyms everywhere, Germans are very fit and love gyms their price is around 50€.

It will be very difficult to find any job without German language skills - thus I advise you to attend a few month of intensive classes in the beginning if you're serious about staying in Germany for the longer term. Such classes are offered by Volkshochschule and similar organisations, where you can reach A1 level within 6 months (full-time studies) for a €600-800 fee - a real bargain!
If you don't have credentials (certificates, relevant work experience), you will face problems finding a job in IT.
And without good German, customer-facing jobs are also out of the question (e.g. waiter, call centre, etc.).
Thus you are probably stuck with typical stop-gap jobs in the beginning, like stacking supermarket shelves, harvest help, etc. These are very hard work for little money, so be prepared! (Only if you're lucky you will get the official minimum wage of €8.50/hour, which translates to €1000/month after tax and compulsory deductions.)

A Cheap Gym is for example  McFit  20€/Month.

Hello,

German language is very important here, specially if you are planning to stay for long time. So I would advice you to take German language course. You can find the list of German language institute on "samtog" website.
As your background is in IT and having 2 years experience will make it a bit easy finding a job.

I also think learning German is critical to getting along here but the IT field might be one of the few exceptions one might still get a job without it - at least to start. That said, just a 2 year program without working experience in the field makes for a rather thin resume.  That you have relatives to help you get started is great.

Many gyms in Stuttgart used to be in the 70 to 100 Euro/month range. Some are cheaper but might not have such extensive facilities - so it depends on what you want. But there are often introductory offers they might be much cheaper. There now seems to be a lot of competition with gyms so I think prices are going lower. Some gyms are now taking the strategy to have a fairly cheap basic membership and make the serious money by charging extra for courses, providing personal trainers etc. Otherwise, many seem to lower the price the longer the contract you sign but it also makes no sense to get a 3 year membership if you don’t know if you are going to stay around or if you decide you don’t like that particular gym. Proximity is also a big factor since even with a low cost, a gym across town that takes 40 minutes and 3 changes of buses to get to is going to be hard to motivate oneself to really use.

Sports groups are also very popular in Germany and can be a fun way to keep fit and also meet people. Some, but not all, of the University sports groups are allowed to take a number of non-students. They will run on a semester basis and are relatively cheap and will be mostly young adults, often very international and all students will know passable English.