Passive - When out of combat for 6 seconds you get a movement speed buff. Makes up for not having a mount.
Z - Mount - You dont get a mount. But you get this awesome teleport to anywhere on the map ability with a cooldown. Make sure to use it to gank, help out teammates, capture objectives, get back to lane for xp, etc.
Q - Hammerang - Sends out a hammer in a line and comes back to you. Similar to Ahri's Q in League of Legends. the hammer damages and slows enemies
W - thunderstorm - Lightning based proximity damage. As long as the target stays within the proximity range they will continue to get hit by the Thunderstorm
E - Barrel Roll - Your escape ability. Jump over walls, close the gap on enemies, escape enemies, etc. etc.
R1 - Shock and Awe, laser like nuke. Does a good amount of damage, takes a second or so to channel. Great for doing lots of damage with good positioning in a team fight or picking off enemy champions running. My usual choice unless I am in a pre-made game.
R2 - Mighty Gust - Pushes enemies away and applies a 60% slow that decays over 3 seconds. I typically only take this ability in pre-made groups otherwise your teammates don't typically know how to position or take advantage of this awesome ultimate. This is great for separating out their diving champions from their carries. Use this, then have your team dive down on their front line. Or the alternative, if your team has a lot of dive, then use this to separate out their front line and focus their carries. The possibilities for controlling the fight are endless.

Top Recommended - Basic Auto Attack Based Build - With this build, you are essentially building an auto attack based champion. If you aren't comfortable with positioning or auto attacking, dont choose this build. A lot of newer players will struggle with this build, since newer players typically play the "ADC" as a caster (League of Legends reference). Most of your damage will come from auto attacks in this build.
Level 1 - You pick up seasoned Marksman at level one in order to stack up your auto attack damage.
Level 4 - I typically take Vampiric Assault. This allows you to stay in lane better by gaining some sustain, then you can save your global mount "Flight" for offensive situations.
Level 7 - Battle Momentum or Secret Weapon - Battle momentum will reduce your cooldowns allowing you to barrel roll more frequently. It isnt the greatest ability but it helps with overall damage output. The other option is to take secret weapon. Although secret weapon is best when paired up with level 1 power throw where your hammerang travels farther so you can get more auto attacks in (Hammerang autoattack build)
Level 10 - Shock and Awe - I will always take this in solo queue, mostly cause I'm trolly and want to dominate. Also because games will be unorganized and you can pick off lots of running heroes to turn the tide.
Level 13 - Giant Killer - Increases your auto attack damage. Nuff said.
Level 16 - Hammer time - stuns enemies after you hit them with your Q. This is important for kiting and the other abilities at 16 dont really fit in with this build. If you are getting dove a lot, then you may want to take Afterburner.
Level 20 - Nexus Frenzy - I use to always take Call of the Wild Hammer because it helps your ultimate and everyone knows their ultimate is the most important thing... right? Lately I've been taking Nexus Frenzy because, well... it owns. With the increased damage from your level 1 ability stacking, sustain from your lvl 4, reducing cooldowns from lvl 7, increased damage based on enemies max health from lvl 13, and your stuns after hitting q from lvl 16, the increased attack speed and range is amazing.

2nd Choice & Also Really Good - "Hammerang Auto Attack Build"
This build is a bit more ability focused than auto attack focused, therefore it is a bit more forgiving for newer players. It is also a very strong build although I still prefer the first build. This build is great for unorganized games where you will be chasing a lot for kills since your Q has ridiculous range. The idea behind it is that you do more damage with your auto attacks while your hammerang is out. Aka, throw it, hit the enemy with it, then light then up with your auto attacks and W while the hammerang is in the air. Use your barrel roll to stay in range of auto attacks for the enemy but outside of their damage range.

Level 1 - Power throw - must have for this build. If you dont take this, dont do this build.
Level 4 - I typically take Vampiric Assault. This allows you to stay in lane better by gaining some sustain, then you can save your global mount "Flight" for offensive situations.
Level 7 - Secret Weapon - This is a must have for this build. Allows you to do more damage while your hammerang is in the air... remember your level 1 ability?
Level 10 - Shock and Awe - I will always take this in solo queue, mostly cause I'm trolly and want to dominate. Also because games will be unorganized and you can pick off lots of running heroes to turn the tide.
Level 13 - Giant Killer - Increases your auto attack damage. Nuff said.
Level 16 - Hammer time - stuns enemies after you hit them with your Q. This is important for kiting and the other abilities at 16 dont really fit in with this build. If you are getting dove a lot, then you may want to take Afterburner.
Level 20 - Nexus Frenzy - This is a must have for this build. You do 80% more damage when your hammerang is out. Now you do 80% more damage with 20% increased attack speed and range. This is a very strong combo, especially with your built-in kite. If your team has any sort of clue on how to play and can keep the enemy from diving you then you will do tons of damage.

Other Choices - "double stacking imma carry this game" and "Caster Falstad"
Double stacking imma carry this game - you are very confident, very trolly, and you probably are very toxic to play with. Have fun. If you play well, this build snowballs and does ridiculous damage.
Caster Falstad You'll do a lot of bursty damage with your spells, but your overall damage will suck. This build is for newer players that are not comfortable playing an auto attack based champion. I dont recommend this build at all unless you are new to MOBA. And if you are new to MOBA, you will probably do better playing a tanky champion to keep you from getting killed when you are out of position. In fact, if you are new to MOBA, dont play Falstad or any other assassin.

My player name is Mattrat3000 and I main Falstad. I find him to be very strong with several different options for how you want to play him. If you want to see Falstad some of these builds in action to get a better understanding of how they work, feel free to check out my stream at http://www.twitch.tv/mattrat3000

Most of my streaming games are strictly Falstad queue