Do I need the gem store?
Some might say NO, but the honest answer is: There are Quality of Life items I wouldn't want to miss since they make the game experience much smoother. Oh, and if you want the fanciest looking skins, you'll need it too.

Are there restrictions for f2p gamers?
f2p accounts can't convert gold to gems and they have an initial 4 day restrictions on gems to gold conversion. They can't upgrade to deluxe versions of DLCs with gems. They can't buy stuff that is connectet to these DLCs - such as wings, mount skins and some seasons of the Living World. Other than that, the items offered in the gem store don't differ between f2p and paid accounts.

How to lift the f2p restrictions?
If you buy at least one DLC you will get rid of restrictions concerning the gem store, plus some affecting your account otherwise. However you still can't buy stuff connected to DLCs you don't own. e.g.: No Path of Fire => No skimmer-mount => No skimmer-mount skins.

Can I buy DLCs with gems?
No

Can I buy Living World seasons with gems?
Yes - if you own the connected DLC

How to get gems?
Press the "o" key or the little lion icon on top left of your screen and the Black Lion Trading Company (called BLT afterwards) window will appear. If you want to spend real world money, you can click on the blue "buy gems" button and select how much money you want to spend.
But there's also an option to convert your earned in-game gold into gems. If you have opened the BLT, there will be three icons on the left side of the window: Gem-Store, Currency-Exchange and Trading-Post. Click on the Currency-Exchange and you'll see how many gems you get for your gold. Don't be discouraged by the conversion-rate. You'll get plenty of gold later ;)
There's a limited third option too: every 5000 achievement points you'll get 400 gems.

Can I buy everything with gems?
Besides the DLCs - which are never sold by the BLT - you can buy everything what the BLT has to offer with gems. This includes the Living World Seasons too. All gems are the same, there's no restriction if you did convert your gold into gems.
Please note: Not all items are available at all times. This is especially true for skins and combo-offerings. You can check gw2 wiki about what's currently available and what's not.

What about sales?
Traditionally GW2 offers two big sales a year. One in March and one in August (since I'm writing this in late Aug 22, this years August sale seems to not happen - maybe they switch to something like black friday in the future, but that's pure speculation.)
Additionally there are also smaller sales every now and then. Most times items offered during sales are beween 20% - 35% cheaper, but sometimes also up to 50% - 75% this big reduction is mostly for things like Living World seasons or Deluxe upgrades to DLCs. Here I would like to mention a particular good DLC upgrade: If PoF deluxe upgrade is reduced by 75% you'll get - amongst other stuff - a character slot and an "invitation to Lily of Elon" for just 600 gems.
So, if you are a new player or don't want to spend a lot of gems, it may be worth to wait a little, but be aware that gold to gems rate usually explodes shortly before and during sales.

What to buy?
Ok here is what I think is the most usefull for new players. If you decide to stay a f2p, this may not be very helpfull. Prices indicated are the normal prices on the gem store, without any sale or discount, please note that some items can also bought at a lower price if bought bundled in a package. And once again this is my personal opinion and must not necessarely be true to you.

Honestly: if your new and willing to pay for it: this steam elite jump start package is an awesome deal :) There are some better harvesting tools, but the ones inside this package will do the job and have an unused glyph slot. For all others here's my list with most usefull stuff:

1. Copper-fed-salvage-o-matic (700 gems)
The thing in GW2 is: Mobs don't drop a lot of gold but items. Now especially in the beginning your inventory will constantly be filled with junk you want to salvage and then sell the materials on the Trading Post. Later on you'll get stacks of unidentified gear you'll want to salvage too. Now ofc you could buy salvage kits from NPC, but honestly: An endless salvage kit is just super useful and in the long run even saves you some gold. IMHO it's really the most usefull item you could buy.

2. Inventory Slots (200 gems)
The inventory struggle is real, so a bigger inventory is just very handy. The problem with this is, they are per character - so if you play with multiple characters, this will get costly quite quick. Please note the character limit on that - sadly you are able to buy more of these than your character can use (check your inventory tab if there are still usable slots on your character before you buy it). Also note you could buy/craft bags with a capacity of up to 32 slots.

3. Endless Gathering Tools (up to 1k gems each, look out for discounts and maybe save your gems untill you can afford to buy them bundled. You could also buy them with black lions statuettes which are awarded for your characters birthday or found in Black Lion Chests)
I really like the convenience of those. But sure you could just buy the normal ones from NPCs - or don't gather materials at all. Guess that's a really personal thing. Also note there are different gathering tools. Some just add a nice animation to it, some will gather extra material each time you use it and some have an unused glyph slot which means you can buy a glyph which then adds the ability to gather extra material each time you use it or make the gathering process faster. At the moment of writing this, the tools with the quickest return on investment are: Volatile Magic Harvesting Tool (pays for itself after 24k uses), Watchwork Minig Pick (38k) and Volatile Magic Logging Tool (69k).
As you can see, it's really more about comfort than about RoI

4. Shared Inventory Slots (1 slot 700gems, 3 slots 1890 , 5 slots 2800)
Bought Endless Gathering Tools, a Salvage-o-matic or even a lounge pass and have multiple characters? Use this to share mentioned-, or any other non soulbound items, between your characters. Oh hell, how I love these. But as stated if you don't play with multiple characters it's completely useless.

5. Storage Expansion (800 gems)
Every Storage Expansion will let you store an additional stack of crafting material.

6. Bank Tab Expansion (600)
Did I already mention the struggle with low inventory space? This will expand your bank by 30 slots.

7. Character Slots (800 gems)
More characters more fun. Or just use them as additional bank, park them on a place with daily lootable chests or at a place you can gather valuable materials easily.

8. One of: Mistlock Sanctuary Passkey / Armistic Bastion Passkey / Thousand Seas Pavillion Passkey (1000 gems)
Having permanent acces to ONE of these is super conveniant, but frankly its absolutely no necessity. Those three will also teleport you back to the exact location you accesed them from, unlike some other lounges.
A little bit special is the "Invitation to Lily of Elon" It is only available if you own PoF and only accesible after you completed the "Sparking the Flame" story chapter of the PoF story. However it can also teleport you back to the location you accesed it by interacting with Rifaah, who is located inside the lounge.

What not to buy
As a rule of thumb: every item that has a one-time use, every item with a limited use-time, buffs (just in case that that one time use thing didn't sink in yet) and Mystic Salvage Kits. I personally dislike the concept of loot chests so I don't recomend buying Black Lions Chest Keys, however: if you like gambling go for it.

Final Thoughts
Added everything together that's a bloody big amount of gems. Well yes it is, but the good thing about GW2 is, you don't need to pay a single penny for gem store items if you don't want to. I personally never felt like I had to grind for gold, it just accumulates naturally. On the other hand, I was playing the game since start and I also don't spend gold/gems on fashion, fusions and other cosmetic stuff. So maybe a new player or someone who is really into the fashion game may have different feelings about that.

I wish you all a nice time in Tyria. Have fun

Heart of Thorns adds some new maps, introduces gliding ability (which is super useful), a new playable class called Revenant and elite specialisations.

Path of Fire adds some new maps, new elite specialisations and introduces mounts.

End of Dragons adds some new maps, new elite specialisations, a new mount and introduces fishing. Additionally if you don't own PoF you get access to the raptor and the springer mount and if you dont own HoT you get acces to gliding. (However I don't have a clue if or how you can lvl up the mounts as well as gliding without the other expansions. So afaik it's just basic acces to it)
If you don't own HoT or PoF you will also get acces to the Revenant class.

Secrets of the Obscure If you don't own Path of Fire this will grant acces to raptor mount and a new way of obtaining skyscale mount. For unlocking skyscale you otherwise have to own PoF and the complete Living World season 4. It also adds new maps, a new combat system, and a new way to make legendary armor.

Since I still think ArenaNet doesn't communicate this well enough: Unless you buy the complete collection you'll have to buy the Living World content via gem store. And you can't play the Living World seasons without owning the correponding DLC.