

Gems are the 'super' currency of Big Farm Homestead. You can use them for premium buildings, skins (future content), market order slots, production storage boxes, skipping time, filling stock shortages, etc. One of the amazing things about this game is they are attainable for free (daily quests, achievements, rewards, etc), especially in the early game. I think it's wise to talk about them early in this guide so you don't make mistakes you will regret and so you will feel confident in your decisions.
I do not work for Homestead. I am just a fan and player. You can play this game anywhere from free to spending a lot. I am a light spender ($2-3 per month). There isn't any space in this game for judging people about their spending habits. I'll be honest, I personally don't care if you spend, or how much you spend. What I do care about is that you get the most for your gems and money.

A side note because of timing: the second construction crew is an amazing package for when you get to level 15+ or if you choose the Flex Farming strategy (flipping buildings around to meet demand - Guide F). It is available when you start, and 36-48 hours after you start. We have not seen it since and are waiting for the store to open and it to be offered again.
Remember that in this game, you play against players with similar skill. If you increase your farms output or tournament scores with gems, you will need to continue to do so, or be mentally prepared take a step back if you reduce spending. Premium buildings (houses) upgrade with gems. The Sundale Premium House starts at 375 gems, but level 2 quickly jumps to 2,500. Having a level 1 premium building sitting around that you aren't willing to upgrade is a waste of space, so know the cost before you invest in premium buildings.

At level 4, you have gems, and the only option is speeding things up (tutorial) or a Premium House. I would wait and save them up. I think the 2 best purchases are the market contract slots:
If you are looking to invest more into your farm, Premium houses are good. They give more workers and require less happiness, giving you more space to work with. Just consider the cost to upgrade them as far as they need to be. I can't comment further until I collect the costs.
Guild wealth (Level 12), used for Guild improvements, can be purchased with Gems.
High performance farmers may find a continous supply of gems helpful to speed up the bottle neck processes to further increase the output of their farm. An example would be if the silo can support 6 groves, speeding it up will allow for more groves, and is cheaper than buying tomatoes with gems.
If you only have one choice, market slots will have the largest impact on your game. I personally go Slot 7, 1 storage box per island, and then Slot 8, followed by more boxes.
There are so many packages in the game that come with gems and other goodies. Generally, you get a discount in these combination packages, and it's up to you to determine the value of the goodies. I'm going to focus on the 3 main packages to get gems. The first is just straight up buying gems (Gem Shop). It is not a great deal, relative to other offers. Every week, there is a piggy bank that you fill with gems by sending contracts, barges and trains. These are 'virtual' gems and don't actually exist unless you buy the piggy bank. At low levels (see adaptive pricing next), you get twice as many gems per $ spent.
The game uses adaptive pricing, which means if you buy a package, the next ones offered will be larger, with a higher price point. It takes multiple weeks for the packages (size and price) to come back down. Early chatter is that the large piggy banks are not a great value.
The best package in the game right now is the Growth Fund: Gems I. It gives you gems as you pass level milestones. It is almost 5x more gems per $, compared to the straight gem packages.
They changed the piggy back to give half the gems, coins and a coupon. Check what the Piggy Bank is actually offering before you buy it. For $2USD, it should be 500 gems.