When considering switching to kegging the equipment you will need is a dedicated refrigerator (Yes a picnic tap will work the same way a cooler keeps your food cold when camping, but no one lives their everyday lives out of a cooler. Check out craigslist. Someone is always selling an old frig), CO2 tank, CO2 regulator (I strongly recommend a dual head because you will want to use your CO2 tank to purge and fill kegs while at the same time pressurizing your keg-orator), shank, faucet with handle, and various connectors, clamps, lube, and hoses. Now realistically, I would plan a multi tap set up because after you've had one for a bit you'll want more and you'll probably want to be drinking one keg while carbing another so I would highly recommend a multi head secondary regulator as well. Also, if you care about maintaining the proper level of carbonation throughout the serving phase across varying styles of beer a secondary regulator is a must. More on that later.
Kegging involves performing two separate but related functions, carbonating and serving. Right now your carbonation is done by putting a specific amount of sugar in your bottles/bottling bucket, capping them and waiting x weeks. As for serving, just bust off that cap and you're good to go. With kegs the carbonation and serving functions get a bit more complicated.
Forced carbonation is simple in theory, apply x amount of psi and over ? days you get fizzy beer. In real world terms when wanting to hit a specific CO2 volume it's a bit more complicated. If landing on a specific amount CO2 is not really a concern then crank the pressure to 30+psi and shake for 10-15 minutes and skip to the next paragraph. But if you have questions like, how do I hit my target CO2 volume, and when do I know it's finished, read on. For solutions to these questions I looked at how the pros do it. When real money is on the line people are less apt to guess. The answer is using a carbonation stone (0.5 micron), a gauge that measures the pressure in the head space and a rotameter to measure actual gas flow into the keg. Here's the link that explains how to do this.
https://chme.nmsu.edu/files/2016/06/Brite-tank-carbonation.pdf
To serve the beer from your kegs the concept again is simple, pump CO2 into the keg, open the faucet, and out pours a beautiful glass of beer. But in reality, getting that perfect pour is a delicate balance involving temperature, pressure, length and inner dimension of the beer line, and gravity. For temperature you need to keep your kegs and beer lines below 40F. For pressure you need to know the CO2 volume for the beer, the temperature of the beer and then use this chart to set the pressure.
http://www.kegerators.com/carbonation-table.php
The point of matching your serving pressure to the beer's CO2 volume pressure is so the beer in the keg maintains the proper level of bubbly from the first pour to the last. If you have different beers on tap that have significantly different CO2 volumes you will need to set each beer's serving pressure to a different level. Which is why a multi head secondary regulator is so useful. The final variable is your beer lines. A kit with a generic 3 feet of line is next to useless. You will be cutting your beer lines to length when you know what your serving pressure is, the lines inner diameter, and how far above or below the center of the keg the faucet is. Then use this calculator to give you the length.
http://www.kegerators.com/beer-line-calculator.php
Put all this together and baring leaks, you'll have a fully functioning keg system and you'll wonder why you didn't do this sooner.