By the way, doing self-employment IT support?

I would like to build on the side for private users a kind of self-employed IT support. Offer me, that I have all sorts of problems around PC, notebook, tablet, mobile phone, etc. Angucke and help.
Can someone give me tips on where I can best advertise and what I have to consider?
If you make a bspw. For example, an ad on eBay classifieds, it will be deleted after 1-2 days again because it is seen there directly as a trade.

Of course Ebay deletes commercial ads, if you do not declare them as a trade ad.
What you have in front is now a commercial activity which must be registered by you to the responsible office, you must also report this activity to the tax office and apply for a questionnaire for tax collection.
Depending on how many sales you make and if you offer your service predominantly you may have / or it is worthwhile to show and pay VAT. Little turnover (previous year max 17.500, - € / current year not more than 55.000, - €) can be compensated by the small business regulation. Use §19 UStG. If you do a lot for companies, it might be wiser to do without this rule, as you can offset taxes on all business expenses.

For your activity, you must also offer warranty, as would perhaps a corresponding insurance is not wrong, otherwise it will very quickly become very expensive.
Example:
You make something of an entrepreneur on his service laptop but by carelessness (can always happen) comes a virus on the laptop. This device comes into the store and connects to the internal network and the virus spreads and lame everything there. It can be determined the laptop and where this "processed" because you have written a Rechnug.
Depending on the size of the company, it can sometimes cost a 5-digit sum or significantly more.

You need to open a commercial account with Google and deposit your phone number, company name and address. This serves as an imprint. Without this account your ads will be deleted again and again.

Just to earn something on the side is not so easy, also all income must be taxed to the tax office with the income tax return. How depends on the type of profit determination.

As this is a commercial activity, it is no wonder that EBay clears the ad.

That may be better for you too, because otherwise you can use it to quickly get a warning.

I just wanted to do it for the private user. But thanks for your detailed answer.