When trying to get rid of a sinus infection, is it better to blow boogers out or suck them in?

Flush it out. The answer is neither. The best way is to perform a saltwater flush of the nasal passages. This will gently remove the mucus/phleghm without the need for blowing with significant pressure. There are alot of different ones you can buy that are cheap. Neilmed sinus rinse is the one my patients like the most(no financial ties).
Blow. Use a humidifier and then get a bottle of otc nasal saline and liberally use up in the nose. Blow gently and often to clear the nasal passages. Limited use of decongestant sprays (no more than twice a day for 3-4 days-----period) may help to unstop the sinus passages and help them to drain.