anyone know which blog module can do this??
Blog should be the module that appears on the mainpage. People should be able to leave messages that will appear on the mainpage. This should only be done by authorised members. The blog should also allow categories and ONLY
showing blog entries of that category. Example: When you enter the community, you will see all category-entries, but when you click on "Admin-Related", it will show you only those entries that are admin-related.
Each entry should be shown like a blog on blogger or any other blogging software would. It should include the poster, the time, the category and the
text including any images/links/whatever.

Your are asking for basic/standard XOOPS functionality. You can set any module as your "home page" and the news module that comes with the XOOPS core does all of this (and much more--like providing an automatic rss feed). Rename it "blog" under system admin>>modules if it helps.

It shows all posts(most recent first) by default but one of it's standard features is the "topic selector" that only shows articles/posts related the topic you select.
(If you can't see the topic selector, go to news then preferences and choose "yes" beside "Select yes to display navigation box at the top of each news page".)

News performs the other basic functions you mention--and you can control who has access (read articles) and admin(write articles) under System Admin>>Groups.