Work Permit or EU Family member Residence permit

After you married, you can move anywhere in the EU with her.
There is of course some paperwork involved (and family reunion visa are still national, even if you are entitled to one as spouse of an EU citizen). The formalities differ from country to country and you might be required to show sufficient income (to make sure you do not rely on government handouts), living space (rental apartment), local language skills and other things. A family reunion visa normally includes a work permit that allows free labour market access, but you should verify this with the authorities of your chosen country. (If a work permit is not included, and you want to work, you need a national work permit instead.)
Please note that family reunion visa are only valid as long as you live together with your spouse. Thus living apart would not be possible!
Nobody can foresee the situation in the UK after Brexit, so the above may or may not apply there and then.

Thank you.
I already thought of that. My main concern is having back up should incase my Visa is rejected. That's why I was hoping for free visa travel.
Moreover, if my Visa is rejected then I will not be in Hungary as well. So I need to keep my plans of working here, as well as keeping open her choice of moving abroad.

If your visa is rejected, you have to leave the EU.
There is no other option.

Actually that's my fear. Because it's easier to get visas while still within Europe.

In general, what would you suggest I do?

You can either find a job and get a work visa, or marry and move togetheron a family reunion visa, or you attend another course and extend your student visa.
All of these are national visa and allow you only visits of other Schengen countries - there is no option to "move freely within Europe" available to you, short of acquiring an EU citizenship.

Hi Jellea
If you are married to an EU citizen, and you wish to live stay within that country, you need to apply for a residency visa. After you have this visa, and if the UK is still part of Europe you need to get your wife to apply for an Article 10 residence card . This visa card will permit her to invite you to the UK. It sounds double dutch but its the correct way.
Your problem is time.
WWW.gov.uk

Thanks. Time is really the problem. To get article 10 residence card, my wife and I need to live in another EU country, where she is not a national. By this she will be exercising her free movement right.

This process made it difficult.

I thought you ARE living in a EU country?!?

Jellea, I am a bit confused as to where you both are living at the moment, but I wanted to make you aware that family reunification in a non-EU/EU couple can (and most likely will) take 6 months, at least in DK. My wife has been waiting for her residence permit to come out for 4 months.

Good point! It takes 3 - 6 months in Germany and similar times elsewhere too, I assume. And no forecast is possible for UK after Brexit ...

Okay, thanks again for the advice. Since my Visa will still be valid till July. Do you think I can use it to move with her to Ireland ( if not Ireland, then which other options), then apply for residence within Ireland?

Yes, I reside in Hungary right now.

With a Hungarian (national) student visa you cannot move to any other country. You first need a visa for the other country.
If you want to move to Ireland, you should ask on the Ireland forum about family reunion visa there (or you can probably find that information by a simple Google search).

Thanks. I will be doing that shortly