Custom template Index

Hey there everyone. I am new to using wordpress and there are a couple of things i would like to use on mine. First, I would like to have my categories listed with the titles of posts under each category displayed, without the ‘Uncategorized’ category. Also, below the categories list, I’d like an archives display with a list of the months (with years). I don’t need to show each posts title with them but, when a month is clicked the post titles from that month appear. I am putting all this into a custom page.

Thanks!