If you have starter access, you are able to access the current campaign, and the official training campaign. Both are accessible by selecting 'Play' and then the 'Solo' tab.

In the 'Create' section, you are able to access the simple track editor, and the replay editor. You can also create basic car skins with the ingame editor, however you cannot use them.
(edit: I think you can now access advanced editor as well, but that changed since I last had starter)

You are able to access the online servers for Royal, Ranked and Arcade Rooms,.
To find them from the main menu, select 'Play' and then the 'Live' tab.
In Royal and Ranked, you can invite your friend[s] to play as a group.
[will confirm if this supports Steam friends, or only Ubisoft friends]
In Arcade Rooms, there are 2 servers you can join. One of them is a server switching between the current campaign tracks. The other is a server showing maps from selected player-made map sets, which changes map set every hour.
There may be multiple servers for each of these options, so if you want to join a friend I advise you to join their game instead of connecting to the same server option

note that if you do not have paid access, you cannot use ingame chat

If you have standard access, as well as the above options you are able to play any map and join *any* club server.
However, you are not able to access it from the club in which it is created, instead you have to:
For campaigns, you have to search for it in Play - Solo - Library, and for servers in Play - Live - Arcade Rooms.
You are able to join the 'Track of the Day' Live server, and participate in the 3 daily Cup of the Days, as well as playing any Track of the Day map in solo.
Also, you can open any local map in Play - Local - Play a Track

If you have club access, as well as joining servers / campaigns from searching for them, you can join them directly from searching their club in the 'Clubs' tab.
You are also able to create clubs, and servers and campaigns etc. as well as joining clubs to have them appear at the top of your clubs list.
You can pin a club, which makes your default stadium when playing a local map / editor show the decoration from that club.
You can also (from a different club, if you want) select a club tag, which will show in front of your name.

Also, you are able to select a custom skin from a club's skins, as well as upload custom skins in Profile (click your name) - Garage - My skins

Thanks for reading my guide
There is a full list of just what is available for each access tier at https://www.trackmania.com/detailed-offers

If there is anything else you think I should mention, feel free to comment.
(or if you think my wording is weird, and have a suggestion on how to do it better)