Will red bumps show up after you shave your penis ?

It is possible. If you shave closely enough, there is a chance some of the hairs may become a bit ingrown over time. The razor will leave a sharp edge, removing the tapered tip of the hair and this can irritate the follicle as the hair reemerges. As well, there is a higher likelihood of bacteria infiltrating the follicles, and staph infection of the skin is always a remote possibility.
You could consider just trimming it short, but if you do shave the hair completely off, there is a reasonable chance of some irritation as it grows back. It may well itch considerably for a few weeks after the first few days, if your not accustomed to such or do not routinely shave the area.

I also woundered that same questian

I use to shave my pubic hair and yes I have allways gotten red bumps. I never used an electric razor though... there are not really bumps though, they are merely the hairs growing back. Once the hair has started to grow the redness will disappear and the itching will come (Its crazy itchy). I recommend just cutting the hair with scissors, you can get it pretty short and no itching.