(Updated November 2021!)
If you followed my Beginners Guide, you'll know that you should save those gems as much as you can. THEY'RE RARE AND VALUABLE!! The following advice is useful for all players, but even moreso if you aren't wanting to spend money on the game (or very little).
Ok, so what SHOULD you spend your gems on?
Gems are really useful because they can be spent on so many things. Each stage of the game, gems can be used on something incredibly useful. When you first start playing Empires and Puzzles, you need heroes more than anything. 3 star heroes are powerful and 4 and 5 star heroes are just incredible. So you should only spend your early gems on the EPIC HERO SUMMON. It costs 300 gems and you're guaranteed a 3, 4, or 5 star hero from Season 1. For a long, long time your gems should be spent almost SOLELY on 3 to 5 star heroes. The rare exception will be discussed below.
DO NOT SPEND GEMS ON THOSE DAILY PULLS!
For 200 gems, you get a 1 star hero? No no no. Don't do it! Also, higher level troops aren't worth it yet. They will come! Just focus on heroes, heroes, heroes. Not on skipping, not on more food or iron, not on shop items (like battle items or chests of trainer heroes). Those things will come. Just be patient! Save those gems for heroes!!
Another option, though, is to do EVENT SUMMON. These ARE worth the extra cost. Season one heroes you can get from Training Camps (eventually), and they're usually weaker than the average event hero (although there are exceptions). So, yes, go for those harder to find heroes.
Save for Ten Pulls
Once you have 3+ star heroes to level up in each color, you might consider saving for ten pulls (where you spend lots of gems to summon ten heroes at a savings of about 13%).
Some Advice On Which Heroes To Try For
My advice is to save up 3000 gems and use them to get Season 2 (Atlantis) heroes until you have PROTEUS. He is a game changer, making your life so much easier for many, many levels. I suggest doing this over other events or Season 3/4. Get Proteus. He's a 4 star hero that stops heroes from gaining mana. Bringing him with mana potions on events is like cheating. After you have Proteus, go for Season 3/4 heroes or event heroes. At this point, you should have an idea of what colors you need, so if an event doesn't have that color of hero, don't do any event summons. Event heroes are usually the best, but Season 2, 3, and 4 have great heroes, too, with better odds to get something that isn't Season 1. Trust me on Proteus, though.
After You Have A Strong Hero Roster
Once you have a good selection of heroes (and this is usually pretty far into the game), you start having some more freedom with your gems. At this point, you have to figure out what you feel is missing the most. Here are the most common to the most rare uses of gems that aren't summoning heroes. Use sparingly!!
Buying ascension items: These are almost as valuable as heroes. After all, if you can't max out your best heroes, they aren't really that powerful, are they? Once in a while, you'll be offered a deal for some ascension items for gems (usually two 3 star items for 500 gems). If you really need the items, this can be gems well spent. Still, save this for when you have a top hero that's stuck. Getting those top heroes should be your priority.
Mana troops: If you need top troops, you could start doing ten pulls for 4 star troops that give mana (4 star troops that give critical hit just aren't worth it, for the most part. Go for those mana troops!). Try to save your Epic Troop Tokens for the Ninja Tower event, where you can get Ninja Troops. They are an extra 4s troop that doesn't take away from your chances to get the normal 4s troops. It's just better chances to get a 4s! Nice!
Ascension items: Once you have a 23rd level Stronghold, you can build an Alchemy Lab, and once you have that completely maxed out, you can pay 200 gems to turn a 4 star ascension item into a different 4 star ascension item. This is worth it if you have good heroes waiting to hit 4th ascension. This is very high level, though.
Emblems: Some heroes are just so good that they need to have emblems put on them. Personally, I tend to spread my emblems out, but if you have a great hero, there are occasionally deals to turn gems into emblems. I've never done it, but I can see why people would.
Finishing a level: Ok, RARELY it's worth it to spend the 75gems to revive your dead heroes and beat a level, but this is VERY RARE! Just save your gems and use more world energy. Did you use a lot of battle items? Those are just food and iron. If your heroes aren't surviving, you should probably just be patient until your heroes have more levels (or you have better heroes). And you won't get more heroes if you're wasting gems trying to beat a single level.
Staying in the tournament: The rewards for a higher place in the tournament is rarely worth it. It's just not worth the gems. I will likely never, ever do this. HUGE EDIT!! THIS HAS CHANGED!! They now give AETHER (which is used for limit breaking above a hero's top level) for a good standing in the tournament. It IS worth 75 gems to get into higher placement, once you have a good roster of heroes. I am now using 75 gems any tournament that bad luck has kicked me out early. This often puts me in the top 5% or 1%, where you get bonus Aether. Totally worth it if you think you can get a level or two higher in the standings. This will be much later in your game experience, though. Get those heroes first!
THE GEM SPENDING EXCEPTION!!
Yes, there's always an exception. If you follow the above advice you'll be fine, but there is an important exception to the ONLY USE GEMS FOR HEROES rule: skipping ahead on chests (I wrote a whole tip on this). I know, I know... just be patient, right? Well, skipping on chests is much, much cheaper than for almost anything else, and, more importantly, it gets you MORE in the long run. You see, chests occasionally are RARE chests. This is probably your most likely source for those rare ascension items, so you want as many of them as possible. But you only get a few skips a day. You should only really start doing this once you can easily beat all the stages for the rare chests (i.e. you can easily do Season 1: Province 13). What are those best rare chest stages? CLICK HERE!
My Chest Skipping Technique: When my monster chest gets to 6hrs and 17mins, it costs 10gems to skip it, and that's usually when I do it. Maybe if I'm about to go to bed, I'll deviate a bit from that, but that's what I do. If you want the exact time breakdown, you can read my entire tip on skipping time for chests here.
That's my advice for spending gems: SAVE THEM FOR HEROES!! (or chest efficiency, after you are on Season 1 Province 13). Good luck with those hero summons!