I'm fairly new to the league and don't always have a lot of time between chores and family time. I want to get better and try new characters and I'm enjoying Anivia (who I've bought) but want to try other roles, specifically tank and assassin. Other than just rolling through the random line up is there any advise or certain champions I should try?
It is hard to recommend a tank without knowing what you are looking for. Are you looking to win your games? I can not possibly recommend a better tank champ for winning games then Nasus. You can add me in the game client if you want more specific advice on what tank champ to play and how to play the champ.
If you like assassins and tanks (those are my favorite), play
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, and etc..
They are awesome. If you want a bigger list, I'll gladly make one for ya :D
It's always nice to see other people like and play Anivia. As an Anivia main, if you would like some tips on how to play her/how to build her or about lane match ups then let me know. I'll be glad to help.
The best (and cheapest) thing to do is just try out the champions during their free week. That way you will know if you like them without having to buy them.
I would suggest 
for assassins. They offer simple concepts to use and are rarely banned. Ahri is safer in terms of mobility due to her Q and R. Akali can be slippery but people can take you down while your skills are on cooldown with a pink ward. Talon is just fun to use in general, but is the most risky of the three.
I'm not quite use to tanks but 
and 
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Malphite is a great pick in any situation and is easy to get use to. Gnar on the other hand takes some getting use to due to his two different forms. You'll want to get the hang of managing his bar to either bait an enemy or sustain off harrass under turret. Both are capable of dealing decent damage, soaking up damage, and just disrupting the enemy's formation.
edit: Just some simple tips for playing tanks in general. If you initiate in on enemy, don't stand there and aim for kills. What you want to do is slowly kite the enemy back because they are not going to focus on you. They want your carry. So move toward your carry while you're using your abilities/attacks on the enemy and turning your focus to peeling. Basically, just keep getting in their way.
Also, eggnivia is getting the most epic skin ever. http://3.bp.blogspot.com/-1PLt4MW9njI/VWapracL62I/AAAAAAAAqj4/GKwsUE1a-wk/s1600/33343.jpg
