Tax Filing in Germany

You need a tax advisor called a Steuerberater. They will tell you what documents they need and file your tax returns and you will get back any overpaid tax. For employees (but not self-employed) one can also use a salary tax advisement club (Lohnsteuerhilfeverein), which supposedly are a bit cheaper. There are also lawyers that specialize in taxes but would be expensive and usually only needed when there are complicated conflicts with the tax authorities.

Legally, one can also do their own tax returns. And if one is an employee it is not extremely complicated if you have paid someone to do it the first years and then copy what they did using a tax software. But to do it from the beginning, or if one is not good with numbers and hardly speaks German, would NOT be advisable. 

I am setting links to a couple of the salary tax help clubs that I found in the internet although I have no personal experience with either of them – and both seem to be only in German.

https://www.vlh.de/https://www.steuerring.de/

The tax class does not determine how much tax you owe, just how much is automatically deducted every month.
How much you actually owe is calculated based on your yearly tax declkaration - and any difference to the automatic deductions is paid back (or you have to pay more, in case not enough was deducted).
A change of tax class can be applied for once a year by sending a request to your Finanzamt (Ask your tax adviser to do this for you!). You can only join class 3 if your wife is tax-registered in Germany (she gets class 5 then). An alternative is both of you in class 4, which is better if both earn similarly much.
I suppose she was not yet in Germany when you first received your tax class and that's why you got 1, the default for singles. This does not change automatically.

The bottom line is that the German Finazamt refunds any overpaid taxes – and without making a problem of it.

Taxes are a complicated subject at times but from my experience, the German tax authorities do try to be fair within the limits of the laws – unlike in some countries where they seem to virtually terrorize people with crazy claims. I’m sure there are people who might have a different opinion on this subject but I think it is generally correct.

Finazamt  above should read Finanzamt