Here is a short TL:DR of the things I learned from new friends I made on the streets while playing Pokemon Go.
To be respectful of your time:I'll list them from most effective + least obvious to most obvious and will go into more detail about every point after summarising the trick.

Lucky Eggs

Use them when evolving pokemon.

You start out with 2 lucky eggs in your inventory, when you use one they give you double experience points from all sources for 30 minutes. Save them for when you've caught at least 60+ pokemon. Evolving pokemon gives you 500 experience per pokemon, so for example: evolving 10 pokemon during the time the egg is active gives you 500 x 10 x 2 = 10000 Experience (that's enough to go up a few levels in the beginning) You just got 5000 experience for free!

Curve balls

Curveballs give 10 extra experience points.

When catching a pokemon you can swirl your finger around on the pokeball before swiping up to turn it into a curveball. When you catch a pokemon with a curveball you get 10% more experience from the catch.

Hatching eggs

The speed limit for distance to count towards hatching your eggs is about 20 mph ( about 30 km/hour).

Have the pokemon app open when you're on the bicycle, on the moped or you're in the car in pedestrian zones.
When you level up or go to pokestops you sometimes get eggs. You can hatch them if you put them in an incubator (you start with one and you get more later) and you travel the distance they require to hatch. By cycling or driving in a slow moving vehicle you can cover more distance quickly and hatch your eggs faster.

Winning gym battles when you're still lower level

Use bug pokemon in gym fights if you live in the city.

When doing gym battles, try to use pokemon that are good against the pokemon in the gym.
Every pokemon has an type (e.g water) and one type is good against one and bad against another. (It's kind of like rock paper scissors).
Here is a list of all the pokemon types and which types defeat which other types:
http://pokemondb.net/type

I found that leveling up bug pokemon are good if you want to win early gym battles and you live in the city, because some of the most common pokemon in cities are psychic and dark type pokemon (which means they are most commonly used in gyms), and bug pokemon are super effective at beating them.

Please comment if you know of any tricks that we're particularly effective to you and that I might not know about.

Cheers,

ZMB