First, when you're new to shaving you always want to shave with the grain of the hair, ie. in the direction the hair comes out of the follicles. Second, all those red bumps are from the sharp ends of the hair snagging and irritating the skin as it grows up and back out the follicles/pores. Some of them are probably ingrown hairs, which you may have to extract by actually breaking the skin to pull the hair out. For that you'd need either a good sharp pin, or some very fine point scissors (in either case, sterilize the metal with alcohol first).
It'll take you a good month of regular shaving for the irritation to go away so you can shave with little to no red bumps and itching. In the meantime it may help if you apply an after-shave gel, such as something with aloe in it. After the skin is good and used to being shaved, then you can consider shaving across the grain (sideways) for a closer shave, or against the grain for the closest shave. However, even though the skin will be used to it, there will still be a few bumps that show up, and maybe an ingrown hair or two. It does get better with time and persistent shaving. For some reason I don't understand, the scrotum tends to be far tougher for shaving and never gets any irritation or itching, and only very rarely an ingrown hair.
If you really want that baby-butt smoothness but prefer to avoid any itching or red bumps, one thing I can definitely recommend is Magic Shaving Powder. It's a depilatory that can be found in the black men's hair care section of the store, and is a powder that you mix into a paste, apply to the area you want to de-hair, wait about 10 minutes, and then scrape away with something thin, flexible, and with a sharp (not rounded) edge. Something like an old credit/gift card or a thin flexible ruler works well. The stuff in the gold can works better than the stuff in the white can, and smells better too, IMO.