There are several possibilities. The Mango app has had some good feedback - it's free, used to carry rather a lot of advertising but for a small payment you could get rid of those. You can download whole series, flag favourites etc, and you get content from about 8 different sources (this is a few months back, so the choice may have increased since then).
Mangasearcher is also worth looking it - it has more variety, but you have to download each chapter in advance.
To find more information and read reviews of different apps, you can type 'app read manga on tablet' into a search engine and this will take you to a page linking you to different user forums. There you can read other people's views in more detail, and even join the forum to ask more questions if you like. Hope this helps.