This guide is meant for Bronze and Silver players. I have no idea if it is viable at higher levels because I don't play in them...yet.

Please vote/rate/comment about the guide to help others and myself.

Your pulls have to be on-point, and they have to be on the people who are most dangerous to your team at the time. Become a pulling GOD by practicing... a LOT. Learn to pull past minions. Learn to "thread the needle" on pulls. Learn the exact max range so it becomes a "feeling." Learn when to use your pull to disengage and escape. Learn to pull people away from your ADC and do hard engages when a jungler comes down to gank.
Make sure you're switching targets with each auto-attack in team fights, and peeling as hard as you can for your ADC/other squishies.
Your shield lasts 10 seconds (unless enough damage is dealt). That's one of the longest shields in the game, if not the longest (not counting Malphite's passive). Take advantage of the time you have. I'm not going to write out all the ways, so just use your noodle and think about it.

What you need to know is that Nautilus is a tanky mother f**ker with crazy amounts of sustain and damage. You're a godd*mn boss, so act like one. Engage like one. Go ballz deep like one. Be brutal as f*ck, so your opponents want to ban him next game.

Engage your enemy and use the following order of skills:
Q to get in there, and pull someone out. (Save this if you don't need it to engage, and then use it later to keep your prey from escaping.)
AUTO ATTACK your prey. NOTE: The next two skills should be clicked IMMEDIATELY after you auto the target so they can't escape.
E to slow anyone who has come nearby and keep your prey from leaving.
W to shield yourself for a bazillion health.
Use your R to peel for your ADC/APC. Target someone who is too far away for an easy auto-attack. That's right - knock up some a$$hat who thinks he can reach your ADC/APC without your permission. Tell that motherf*cker "nope" as your R knocks him, and everyone else along the way, up into the air for some more hard CC action!
AUTO ATTACK another person.
E again to control the battlefield.
NOTE: Every situation is different. This is about 90% of what I do, though, and it is incredibly effective in this elo.

Ban anyone with a hard pull or stun. They will make your lane-life miserable. Choose the champions you're the worst against.