You can access the discover section by clicking on the the number sign (also known as a hashtag symbol.) There are a few sections including tweets which shows the best tweets tailored to you by twitters special algorithms (mathematical formulas.) Activity shows you what people you follow on twitter have done recently such as who they followed, retweeted or favourited. Who to follow gives you suggestions for new people to follow based on who you follow already.
Lists allow you to group together people you follow on twitter and then see a timeline of only those people. For example you can add all local people, family and friends, book lovers, members of a certain group etc and then see a timeline for that list. You can make a list public or private and you can follow the public lists of other users. To access lists, go to Me and then click more>lists.
Click on the list name to access the list timeline or see the people included in the list. You can choose to follow any of the member directly from the list page. This can be a good way to find new people to follow. Find a list of twitter users that you would want to follow (bloggers, bookworms, artists, etc) and then follow a bunch of users included in that list. There doesn’t seem to be a way to search for lists but check your trusted followers to see what lists they might have.