Fizz is a capable roamer due to his fast walkspeed and high mobility. This build allows Fizz to guard river, help out lanes, gank overextended lanes, and ultimately win the match.

Quints:
Gp10 Quints are a must have while roaming. You can switch this out with Movement Speed Quints, but it'll hurt your gold income quite a bit.

Marks:
Magipen Marks are the way to go, to get that high damage output.

Seals:
Scalable AP Seals are ideal.

Glyphs:
Flat AP Glyphs to give that extra oomph in your early game.

I'm going to briefly go over the most important masteries for this build.

Mana and Mana-Regen (Expanded Mind and Meditation):
As a roamer you want to be able to stay out on the playing field a lot longer.

Movement Speed (Swiftness):
Of course, as a roamer you want to grab as much movement speed as possible.

Gold (Greed and Wealth):
You won't be farming as hard, but your gold income will be on par with your teammates who are laning. The extra gold at the beginning lets you grab that ward at the start of the game, along with your boots and health pots.

Experience (Sage and Awareness):
Since you'll be out of lane more than your teammates, you want to be able to grab that extra experience. Especially with Sage, since you're expected to grab a lot of ***its from you ganks. This mastery will prevent you from getting outleveled as a roamer.

The rest is whatever. Of course you need to take your summoner spell mastery, which both are conveniently available in a single mastery in the Utility tree. I put 4 in the AP mastery under Offense just to get a bit more dmg output early game.

It's quite an unconventional, Utility heavy mastery build. But so many Utility masteries now support the role of the roamer. Feel free to experiment yourself.

Boots, pots, and wards are a good first buy.

You'll want to rush your Mobility boots ASAP to be able to run from lane to lane in a decent amount of time. This makes Fizz an incredibly mobile roamer.

Lucky Pick is your next buy, you the extra GP10 is necessary for you to keep up with the rest of the team, and your opponents.

From here on, you'll want to buy health pots as well as tons of wards. Wards, wards, wards. Ward everything and everywhere. Of course, building Lich Bane is your top priority. So buy the items for your Lich Bane and whatever you have left over goes to wards and pots.

From here on out, you have enough damage output. You don't do as much as a straight AP build, but you do a decent amount. You'll want to continuously spend most of your gold on wards, wards, wards. Wards EVERYTHING.

If the game goes into super late game you'll want take your pick and buy Deathfire Grasp, and a Rabadon's. Whatever you buy, leave a slot for wards, always.

If the opposing team has a jungler, I take Playful/Trickster first. I'll explain why below in the 'Early Game' section. Otherwise, take Seaston Trident, max that out first. Max our Urchin Strike second, and finally max out Playful/Trickster. Always take Chum of the Waters when you can.

At the very start of the game, you have a few options with Fizz:
-If you have a jungler, go ahead and place your ward on the main route of a possible gank. This way your support won't have to waste CV's protecting your jungler.
-Place your ward at where you think the enemy jungler's first camp is going to be.
-Place your ward at your lane if you suspect early gank

Roam Fizz should start at duo lane. Let your partner farm a bit more than you, but don't forget to farm yourself. Dodge CC with Playful/Trickster. Use Urchin Strike and Player/Trickster to harass and get yourself out of trouble.

Now you should have your Boots of Mobility and Lucky Pick. It's time to consider leaving your lane:

-If one of your lanes is in trouble, run over and cover the lane for a bit until things settle.
-If one of your lanes is pushed to your tower, go for the gank.
-Keep warding the map. Keep an eye on how long wards have until they expire and replace them.
-Nobody on your team should be getting ganked if you're doing things right. Call out MIAs and ping on the map when you see a gank being prepared.

Ganking:
Initiate with Chum of the Waters. Don't use Ghost or Flash until your target(s) start getting away. Save something for an exit option if things get too hairy (Playful/Trickster, flash, or ghost).

Teamfights:
Stay in the back, use Chum of the Waters if you see a good opportunity. Turn on Seastone Trident and Urchin Strike their carry. If you start losing health Playful/Trickster yourself out of danger. Rinse and repeat.