I am looking for an SQL database that offers a similar easy way of
designing a database as MS Access does but does not have its bugs and
limitations (for example when handling larger tables/files). I want to
create tables with an assistant or in a table (like in Access) and not
be forced to use SQL Statements to create and define the table. Though
I would like to define SQL Querries.
The database should be fully functional and free to use (I dont want
to spend money in case I dont like it). If you need more information,
let me know.

A lot of people like phpmyAdmin.  You can download it from
sourceforge.net.

You still have to run and install a SQL server, but you can use
phpmyadmin to do queries and make tables.  If you have access to a
webhost, they often include 1 database, and sometimes a phpmyadmin
control panel.

SatyenS

Thanks for the suggestion. My Host has phpmyadmin and mysql. I tried
it, and it lets me create tables. The disadvantage to my opinion is
that it is slow even with a 2MB/s line, and it does not give me a
table view to enter and edit values.

If you like the MS Access frontend, you might try using it with an
MSDE backend.  While MS Access is not free, it sounds like you may
already have it, and the MSDE is a connection-limited version of SQL
Server that can be used in applications without any licensing fee:

http://msdn.microsoft.com/library/default.asp?url=/library/en-us/dnacc2k2/html/odc_msdeintro.asp

regards, mathtalk-ga