Listing posts in categories

I’ve been searching this all night and can’t find a clear solution and hope someone here can point me in the right direction if not give me an idiots guide to…

I have tried a couple of plugins but still can’t work it out..

What I’d like to do seems simple ? short description of all posts by date are on the home page (index) then if I click blog I want to have a summary list of the posts that are in the blog category and click articles a list summary of the posts that are listed as articles etc ..

Just checking the box for categories in Add New Post doesn’t seem to do it..

Can someone let me know the best way of achieving this and how to go about it… ?

ps: also I’d like to have a next page button when the length of the page gets too long or if 2 blog pages are related..