No goals require you to spend gems to progress the storyline. A building gem purchase might be the option at the end of certain goals, but is not required to progress.
Take your time, there honestly isn't any rush. There are a couple of Baron market requests at the beginning which are timed and if you don't finish and you've loaded some goods, you lose your resources, but apart from these couple, everything is on your schedule.
We all play this game differently and there is absolutely strategy and prioritisation needed. Find what suits you and is relaxing. Just because there is a goal in your book doesn't mean you have to do it straight away. You can wait and just build up blocks, beams and glue for as long as you want before proceeding.