"Popping" as in 'goes bang?' or just that it goes flat? Front wheel, back wheel, or both?
If they're just going flat, you could have a piece of debris lodged in the tire casing that keeps puncturing the tubes. Wipe the inside of the tire with a cotton ball, to see if anything snags.
Yes, you need to close the nut on a presta valve. You don't need to reef it down, but it should be snug, as it's the only thing holding the valve closed, other than air pressure.
Also get a floor pump with a gauge; even an inexpensive one should do fine, you need some kind of repeatability, especially if you want to ride skinny tires.
As a bigger rider, (I'm~220, myself) you need to make sure you have enough air in the tire to support your weight, and prevent pinch-flats. Smaller tires require higher pressure for a given rider weight, so you'll have to keep those 25s pumped up pretty hard. For general-purpose road riding, I like 28-32mm, for that extra air volume bb