I’m an American student looking to visit my family in the UK this summer prior to starting my course in September. My course is less than 6 months so I do no need a student visa, simply just enter with my student documents a week before my course begins. My question is if I could visit my family for the summer (approximately 2.5 months) on a tourist visa, which grants me a 6 month stay. I would leave the UK for approximately a month to visit some other family in Europe before returning and reentering now with my student documents a week before my course. Will this be a problem?
The rules permit you to do what you want. There's more detail in this answer
However, bear in mind that flights are limited at present, Schengen area borders are closed to US travellers, and non-essential travel is severely restricted in Britain.
This might change before you intend to travel, but at least one EU country is talking about maintaining the border closure until September (Source).
You would do well to keep your travel plans flexible, and perhaps take your holiday after your course.