Replacing the faucet on your kitchen sink...... Take AJ's advice. If you are skinny enough to get inside the cabinet beneath your sink, disconnect the waste pipe at the 'P'-trap, shut off the water at the stops (water supply valves), disconnect the supplies (copper tubing supply lines from faucets to stops)...... Disconnect these at the stops, unplug any wires from a disposal if you have one. Here, they are required to plug into an electrical outlet on the wall beneath the sink. If its hard wired, you'll need to find the correct breaker and switch it off at your breaker box.
Disconnect any drain lines from your dishwasher if you have one. These should connect to the vertical drain pipe between where it connects to your sink and join to it above the 'P'-trap if its piped correctly.
Then loosen the screws and hold downs holding your sink to the counter top (once loose, these should pivot out of the way beneath the sink rim, no need to remove them altogether in most cases) and pull it all out in 1 piece. You may need to work with it a little if someone used RTV or other sealants rather than plumber's putty to seal around the sink where it sits over the hole in your counter top, but with a little finessing and a few words you wouldn't want your wife and kids to hear, it should lift out from the top.
Then you can trim it out (change out the faucet and cut some new copper supply lines an inch or so longer than the ones the old faucet had). Always a good idea to use a little plumber's putty under the faucet when replacing it so it has a water tight seal where it joins to your sink.
Before setting your sink back in the hole, wipe the area around the hole clean and dry. Now roll some plumber's putty out on the counter like you would roll dough (make sure counter top is clean and dry first). You want to take some out of the bucket and roll it into balls about 2" in diameter. Then roll them out like dough until they are about 3/4 of an inch in diameter and several inches long. Diameter matters, length doesn't. Place them around the hole, overlapping each rolled piece about an inch.
Once done, set your sink back in place, hook everything back up, tighten the sink hold downs and you are good to go. Use pipe dope (thread sealant) sparingly on your pipe joints where reconnecting. The supplies (soft copper tubing) hanging down from the faucet...... Hand work them into place and trim to length with a small tubing cutter before connecting them to the stops and you are done.
The excess plumbers putty will be pushed out around the perimeter of your sink where it sits on your cabinet. Just roll the excess up in a ball and if still clean, drop it back in your bucket of putty and seal the lid for next time.
Caveat...... If you have a fancy stone countertop instead of formica...... Stone is porous and the oil in plumbers putty can sometimes leave a stain if you roll it out in direct contact with the counter top. In such cases, lay a piece of cardboard or something similar on the counter top and roll your putty out on that.
In my case, I am just too big to fit through the cabinet doors under our kitchen sink. That's why guys like me were running large cast-iron pipe and large iron screw pipe on commercial construction jobs and little guys were setting fixtures. Last time I replaced our sink faucet, I explained to my wife that I was going to have to saw the center divider out that fits between the 2 cabinet doors under our kitchen sink in order to get under there. Still wasn't a lot of fun, but I managed to get it done. After I finished, I cut a couple small pieces of wood for backing and screwed them to the inside of our cabinet at the top and bottom so I could replace the center divider I had to cut out. Then I screwed the center divider in and 20 years later, you can't even tell it was ever cut out.
Regarding the stops (small shutoff valves beneath your sink and other fixtures). Before you go through the hassle and expense of replacing them, wrap the stem area with several turns of cloth torn or cut from an old shop rag, flannel shirt or something similar. Lay another piece under them to catch any oil that drips and soak the wrappings in WD-40 or a good penetrating oil. Allow this to soak in (down the valve stem to the packing under the packing gland nut) for a couple or 3 days. Wouldn't hurt to give the wrapping another spray every evening for good measure. Then unwrap them, wipe off any oil and give them a try.
Once they are freed up, turn the handles all the way closed (don't jam them down hard on the valve seat)...... And open them all the way slowly. Cycle them a couple times all the way, each way. If you have any drips from the valve stem, gently tighten the packing nut a little. It should take up.
Now when you open them for the final time, open them slowly so the pressure doesn't blow the supply tubes off (unless they are pretty old, not many plumbers now of days take the time to flare the copper supply lines like we did many moons ago). Or they may have used some of the new stainless braided supplies that just screw on.
Anyway...... When you open them, do so gently and don't let anyone jamb them down hard in the open position. That's usually how they end up stuck open as yours are. Just open them all the way...... Then turn the valve handle 1/16 turn back towards closing so it isn't jambed open. That is the proper way to open a gate valve and should prevent it from becoming stuck open again.
Now if that doesn't work and you must replace valves that are sweat (soldered) directly onto copper tubing...... Let a plumber who has some experience working with these do the job. There is a certain way you need to prepare the pipe and valve and a certain way you need to solder them on. If that isn't done just as it should be, you can overheat the valve and its no good. Or excess solder can flow back into the valve and screw up the seat so it won't shut fully. Very important...... Make sure they use soft solder...... 40/60 or 50/50 tin/lead solder like what was originally used. The 'lead free' stuff requires a lot more heat like 95/5 tin/antimony and that much heat will ruin the valve.
The other way (if you must replace the valves) would be to remove the old valve (sweat it off) and solder on a fitting having male pipe threads so you can let it cool, put a small amount of pipe dope (thread sealant) on the threads and change over to a stop (valve) that screws on. If I had to do this, I would go that way.
Lowes, Home Depot, Ace Hardware, etc. should have solder and paste flux unless the anti-lead Gestapo in your area has banned them from sale.
Regarding your stops, replacement faucets, etc...... Get them from a real plumbing supply house. The kind with creaky 100 year old wood flooring, cats chasing rats and a well used spittoon by the door. Not from a hardware store or one of the large outlets such as Lowes or Home Depot. There really is a difference in how long they will last, even if you get what appears to be the same part, same brand elsewhere. Its not an old wives tale. Such stores buy in bulk and have the manufacturers provide stock made to their own varying specs. In some cases, you can look at the part numbers on the boxes, compare them with the same part sourced directly from the manufacturer and find the number or letter designator of the retailer they were made for.
Hope this gives you some ideas,
Harry
P.S. >>> Please stay away from PEX tubing if you repipe anything. Use 'real' pipe. I say this not so much as a former pipefitter and plumber weighing the pros and cons of PEX as from the standpoint of knowing that anyone having the skill and experience to competently work with copper tubing or iron screw pipe is much more likely to be a real 'mechanic' and know what they are doing.
