At the start of the game you do need to be very selective about what you upgrade as shards will be the limiting factor. I would avoid upgrading any Common gear, and if you can hold out long enough to find Rare gear, wait until you have those to save on shard usage.
The main source of shards will be opening chests. Higher rarity and higher level chests can drop more shards, which will come with natural progression through the story. You do get shards from scrapping gear - Minor from Common, Lesser from Rare, etc. How many you get is also linked to the level of the item.
Check the store for daily deals as you may be able to spend some gold or gems to buy bundles of shards, and you can view how many you have through the pop up menu in the top right corner of the screen.
Also look to join a Kingdom. These have a daily bazaar which gives another route to purchasing bundles of shards, keys and chest tokens with other resources you earn while playing.
The devs are making balancing changes with each update - the most recent made big changes to evolving gear, raising an item Rare to Epic, using materials that are even harder to find than shards.