What's the best strategy for early play in Pokemon-Go?
If you want to level up fast:
TL;DR:
Save evolutions for lucky egg effects.
Lucky eggs double all of the experience you gain for the 30 minutes after you use them. You get a lucky egg once you reach level 9 and every multiple of 5 thereafter (10, 15, 20, etc.). You can also buy them from the store for Pokécoins (which can be purchased with real money). Being under their effect at all times that you're playing the game will make you level up extremely fast right off the bat, but if you don't want to spend the money and want to level up efficiently the old fashioned way:
- catch every Pokémon you can find, send extra ones to the professor*
- visit Pokéstops
- ignore gyms (for now)
- don't spend candy
- don't spend stardust
- don't evolve (for now)
* Keep enough Pokémon to evolve them:
If you have collected ~50 Pidgey candy you can evolve 4 Pidgeys (12 candy each), so you have to keep 4 Pidgeys in your inventory. (For other Pokémon check how many candy they need to evolve and keep enough of them)
If you have accumulated enough Pokémon to evolve and their candy you can use a lucky egg and evolve them all at once. It's recommended to use cheap Pokémon for this, like Pidgey, Caterpie, etc., which cost 12 candy to evolve[1]. Also it is useful to use lure modules and incense with your lucky egg and double the catch XP as well while evolving your zoo.
When you reached a decent level and you start caring about gyms you can then start spending stardust. Don't waste it as it gets more and more expensive to power up Pokémon later. It's not necessarily needed to level up, but keeps the fun factor higher. If you want to power up a few Pokémon make sure to power up Pokémon with higher CP as they need much less stardust to get combat ready.
Edit for clarification
[1] Of course you can keep more expensive ones as well (like Rattata) and evolve them with the egg effect enabled. If you're really running out of space by that time make sure to keep the ones, where you don't have the evolved form in your Pokédex, yet, in order to also get the "first catch" bonus x2.
Levels 1-4
When you start out, you only have three EXP sources - catching Pokemon, hatching Pokemon and spinning Pokestops.
Catching Pokemon
At the start, catching Pokemon is very profitable - you get 100 EXP for catching, and 500 EXP if the Pokemon isn't already in your Pokedex. You can earn 10 bonus EXP if you spin the ball before throwing it, and more bonus EXP if you can hit inside the shrinking colored circle.
But all these bonuses are unreliable - most of your captures will be regular captures that give 100 EXP.
You can also evolve Pokemon if you have enough candy but don't do that until you have Lucky Eggs (see below).
Spinning Pokestops
Spinning Pokestops is a much better source of EXP. They give 50 EXP every time and refresh every five minutes. So plot a route that takes you past a good amount of Pokestops, and is about 5 minutes long. If you live somewhere like Manhattan, almost every street corner has a Pokestop and streets are 80m apart, so a regular walking pace (1.4 meters/sec) will have you hitting about one Pokestop per minute. Hop on a bike, and you can double or triple that number easy, but even without that, you're earning 3000 EXP per hour.
Of course, this cuts into your Candy supply, so stop and catch Pokemon you want, and Pidgey, Caterpie, and Weedle for evolving later.
Hatching eggs
You get one incubator to start, and can find eggs by spinning Pokestops. Every time you hatch something you get 200 EXP, plus the 500 bonus if it's new to you. Focus on the 2km eggs because they will be easy to hatch, and eggs you get now will have low CP Pokemon. Average walking speed is 5km/hr, so you can do two and a half eggs per hour of walking for 200 XP apiece. That's 500XP per hour on top of your 3000 from Pokestops.
Accounting for the 500 EXP bonuses, you should reach level 5 (10,000 EXP total) in about two hours.
Levels 5-8
You earn an Incense for reaching level 5, and another at level 7. You also get an egg incubator at level 6 and a lure module at level 8. You also earn the right to battle gyms.
Incense
Incense burns for 30 minutes, and spawns a Pokemon every 5 minutes (if staying still) or every 200 meters (if moving). Remember that average walking speed from before? In that half-hour, it will take you 2500 meters, resulting in an expected 12 bonus Pokemon encountered, or 1200 EXP per incense.
Hatching
Don't use your new Incubator on 2km eggs, because it only has 3 uses. Your XP gain will be slower, but you should make up for that when you hatch the rarer Pokemon inside 5km and 10km eggs.
Gyms
Don't bother with them. You get decent EXP for defeating Pokemon, but it takes too long and you're too weak right now.
Lure Module
Plant your lure modules on the Pokestop you plan to pass every 5 minutes on your grinding circuit.
Levels 9+
Level 9 is when you get your first free Lucky Egg, and can start doing mass evolutions of Pokemon with low candy costs. Pidgey, Caterpie, and Weedle cost 12 candy to evolve each, making them excellent candidates. You will also have Pokemon with higher costs that you can evolve during these windows to complete your Pokedex and get powerful Gym battlers.
Because each evolution has a fairly long animation, you can only do so many while your Lucky Egg is active. You have time for around 60-70 evolutions in the half-hour that the Lucky Egg is active, so make sure you have that many Pokemon ready to evolve. To make it easier, add all Pokemon you will evolve to Favorites (tap the star in the top right of their profile), and then remove them from Favorites once you've evolved them.
After finishing your Lucky Egg, send your newly evolved Pidgeotto, Kakuna, and Metapod to the Professor for more candy. Don't do it during the Lucky Egg window, it's a waste of time.
A great tip I got from my cousin is to do all your evolving while the lucky egg is on. All the +500 XP bonuses become +1000 XP.