I also have a 2 year old, and he often has a runny nose (since he started daycare). He cannot blow his nose, and those blue rubber bulbs don't work well. What works best for us is to spray his nose with saline spray (such as Simple Saline, which stays sterile) to soften the mucus and then suck everything out with a battery operated nasal aspirator. It is much better at cleaning the nose than just wiping.
I also sometimes wash his nose at the sink with warm water and get as much as I can out. Warm water softens his nose so it is easier to clean.