Access is not a real database, it might work, but it was designed for one user. There are several good databases that are free. (for some definition of good) MySql, and PostgreeQL come to mind, but I know there are at least 2 more full featured databases out there that you can get for free. Not to mention the pay ones that are better.
Learn database design before you start. Beginners often make mistakes like not "normalizing" their database. It is easy to do once you understand what is going on, but if it isn't pointed out you won't do it. A little (well a lot, really) of up front work will really save you down the road. For playing it won't matter, but if you ever want to do something more you will need it. Fixing a badly design database is hard.
If this is for work you should really consider hiring an expert. There are little pitfalls that a good expert will avoid that you will hit. (I am not such an expert) Some of those pitfalls mean the different between one user at a time on a powerful machine, to good for thousands of users at once on some old machine.
As for languages, you could learn ASP, PHP, java, python, C#, and many more. Each has advantages and disadvantages. A good programer can learn any of the above in less time than it takes you to figure out that when he started he didn't know it. (Mastery can take years of course) Don't worry about the individual language so much as learning how to program.
If you want to learn: don't learn one thing. learn some python, then jump to ASP (which is nothing like python and covers a completely different area), once you know that jump to SQL (which is different again). The language is just a tool - I can drive screws with a hammer if I must (and I have done so), but you are much better off knowing how to switch to a screwdriver.
Note the list I suggested above. You can substitute ruby, java, REXX, perl, or VB for python and you essentially get the same results. (I don't reccomend perl or VB but that is a different matter)
For ASP you could use PHP as well, or one of several other packages I've never looked at. Many just cover the things they do well with Java or something from the first list, which might or might not be better for you.
SQL is the language of Databases. Every real database you encounter (I'm not sure about access, but I think that too even though it isn't "real") speaks SQL in some form.
Remember design is what counts. If you can't create a good design you will suddenly find yourself spending most of your time maintaining this mess (and not doing whatever other work you do, perhaps not even getting a promotion because if you are promoted who will maintain it). Eventally you quit your job for something with less stress, like air traffic control. Seriously though, bad design in a toy that suddenly gets put into production can come back to haunt you for years. So learn the design even more than the programing.