What should I check first in my pool after heavy rain?

Test pH and total alkalinity first. Rainwater is acidic (pH 5-6) and dilutes everything, so get pH back to 7.4-7.6 and alkalinity to 80-120 ppm. After that, shock the pool and run the pump for at least 8 hours. If you got a foot of rain, drain some water to keep the skimmer working.