Regarding My career path
The answer to could be really broad, but I'll try and keep it brief.

1.

MySQL as a long term technology is a really open-ended question - what do you mean by 'long' term? It's unlikely to go anywhere in the next few years, or even decade, but the market is constantly changing. Feeling too secure in anything probably isn't a good idea.

MySQL is still in the top rankings for RDMSs www.serverwatch.com , http://db-engines.com/en/ranking

So doesn't seem to be going anywhere. The fact it's free, and there's no looking like it's going to stop being supported, means that it's probably fairly stable in its position.

The questions you need to ask yourself are:
Do you have a job, and do you like it?
Do you want to do something else?
Can you learn new skills where you are, and what are the requirements there, what can you get training for?

The tricky thing with MySQL is how it's used 

Lots of companies seem to expect MySQL DBA stuff as just something a really good PHP dev does. In my shop (where I'm a SQL Server DBA) the web team tends to look after the MySQL DBs, it's technically not in my remit (and I only muddle through with them if they need a hand). 

Also, if a company is choosing to go open source, that's usually partly because they don't have a lot of money - making it less likely they'll have funding for a full time MySQL DBA (or at least won't be paying much). 

This article talks about this more (it's a bit out of date, but what it's talking about doesn't seem to have changed). 


Apart from MySQL, what DB should I learn?


Again, this is a bit of an open question: I could be flippant and ask back - what do you want to learn? It all depends on your reasons "Easily getting a job outside" is a bigger question (maybe take up being a woodsman, if working outside is your bag..!).

According to rankings Oracle and SQL Server are still right at the top with MySQL, so if it's just number of potential clients is your question - then they're the answer. Problem is, that's not so straightforward. Both of those markets already have lots of established DBAs - that's not to put you off, it's perfectly possible to start in the career, but it takes work. You need proven experience of systems in order to get jobs looking after them.

If you want to stick in the open source sphere, then investigate other open source systems people are using. Are you familiar with MariaDB? Since the split from MySQL it has been becoming increasingly more popular, and there are increasingly more reasons people are switching. Being already familiar with MySQL, this should be easier for you. 

My personal opinion

Familiarising yourself with other RDMSs is always a good idea, but focus on something. Its rare a shop only has one RDMS and no others, even if they are legacy, or have other DBAs, it's always good to be able to help out and make yourself more invaluable - you can also learn more on the job.

I also think the question is not what RDMSs you should learn, but what other skills do you have, learning associated skills helps even more getting a doing a DBA job.

If you're sticking with MySQL or MariaDB, what's your PHP like? 

Having a scripting language is always useful - 
If you want to go into SQL Server (or use anything on Windows), consider learning PowerShell.

Choose a programming language to learn - what will depend on what area you want to go into (as a SQL Server DBA, I'm always trying to improve my .net and C#).

Having good project management skills/qualifications is another string to your bow.

Really, there's lots of things you can do, but there's no quick answers. You've got to research and find out what suits you.

Sorry... didn't keep it brief.