Another +1 for the .22 rimfire to start. Learn your safety procedures and get your basics down, learn from a good instructor and be patient. It'll pay dividends in the long run.
First purchase I would suggest a .22LR or .22WMR
rifle, preferably bolt action
. Personally I really like the WMR, although ammo is a bit more pricey ($0.15-0.18/round as opposed to $0.03-0.10/round). The .22WMR is effective on small animals out to about 125-150 yards, and would kill a deer with a headshot at 75 yards, maybe 100-125 if properly placed. I've achieved some relatively tight groups at 200-250 yards off a rest as well, although the energy delivered at that range is just enough to put a hole in the target so it's academic, but fun. The standard .22LR is at the tail end of it's ballistics at 115-125 yards. (at least, that's my understanding, if there's an expert around feel free to correct me)
My .22WMR is a Marlin 983S bolt action
. Stainless bull barrel, 14rd tube magazine, iron sights but will accept a scope. Laminated stock but it's pretty and durable. Paid $200 cash and it came with a sling, 4x30 scope, bore snake, and about 150rounds of ammo. You can shoot it all day, from a rest or offhand, and have a blast without beating up your shoulder. There are a ton of .22LR options, Ruger 10/22 semiauto is probably the most popular. Marlin makes a Papoose survival rifle that is really slick, it breaks down and can be carried in a backpack. Worth checking out.
Pistol, I suggest a Ruger Mark III Target .22LR
. There are others that are just as good, but I like that one personally. Adjustable rear sight and bull barrel. Under $500 new with 4 mags, cleaning rod and case.
Whatever you buy, try to invest in professional instruction
beyond anything else. Take a safety course and pay attention. If possible, find an instructor who is NRA certified, or shot/shoots competitively.
Regarding your law enforcement aspirations, quite frankly firearms proficiency, while important, shouldn't be your primary focus as you prepare for that career. Education and physical fitness should be high on your list. Run, do pushups/situps/pullups, work on your quickness and agility. Google "Stew Smith" and download some of his workouts.