How to get gems?

No yeah it is exactly that, people have multiple alt accounts for farming cards that they then use to craft badges during the sale as each badge crafted drops an event card. And you can also get one free card per day per account by doing the discovery queue.

And yes, the only ways to get gems is either trashing items or buying sacks of gems from the market. There's some backgrounds and emoticons that can be trashed for 100 gems tho, those are a much quicker way to accrue gems.

buying sacks of gems from the market. There's some backgrounds and emoticons that can be trashed for 100 gems tho, those are a much quicker way to accrue gems.

Either of those options is how I go about it. Never cared for the idea of farming. I like to get the cards as I actually play a game, and then I'll craft the badge after I beat said game. I find that to be fun; just sort of an extra bonus/final act.

Recycling backgrounds and emoticons really can make the gems add up fast. Lots are worth 40-100. That's probably what I would recommend for you, where it sounds like you don't want to mess around with farming. If you want more after that, then you can start buying (or trading for) sacks.

Just mentioning the obvious for completion's sake: Obviously, you can buy gems on the market. Which is the fastest way.

Although, if your goal is to make money by selling gems then it is not an option. If it is for creating booster packs, chances are that buying the card directly on market is cheaper as well. There are exceptions. For example when there is no market for a game or if you believe in your magical luck to get shinies.

Other than that: trashing inventory items. However, this is limited by the number of your items ^^'

I'm not sure that turning cards or items into gems is a good idea. From my own rough analysis, there is a loss of value when converting things into gems. Imho, if you can sell the item on the Steam Market, you'll get more value than turning it into gems.

I trade some of my bundle leftovers for gems and TF2 keys. Then sell those on the market in order to buy the cards I'm missing from the marketplace and redeem badges (or buy games straight from the Steam store, but that's another matter).

I used to trade TF2 keys for card sets from bots (which you can check out at steamDOTsupply, not sure about the policy so not linking directly) but with the dropping prices of gems and keys the number of sets I get now makes it much less attractive.

"Seen some people have hundreds of cards from events like the summer one currently active - I wonder how this is possible?"

Most people save the sets they can make until the Summer or Winter sale so they can craft badges then. Crafting a badge any other time drops a coupon, but waiting until a week before the major sales and crafting from then until the end of the sale drops a sale card. This is why you'll see posts mentioning that mystery cards (unrevealed major sale cards) are active a week before a major sale starts. Once the sale starts, the mystery cards show as the sale cards.