The water and drain lines in the above picture have been replaced recently. I would like to keep as much of that in place as possible.

I just want to remove the sink, faucets, and garbage disposal. Any helpful advice is appreciated.

I will have help in replacing everything, just want to get the removal done myself.

First unplug the disposal from the wall or switch the breaker off. put a tub under the pipes, turn the "shut off" valves full right to which will stop the water supply, disconnect the supply lines from the valve at the wall and unscrew the drain close to the wall.  There will be water in the "p trap" You can disassemble when you get it out of the cabinet.

Easy peasy.  

1.  Get a bucket or a fairly deep rectangular pan to catch drips.  Have some paper towels and/or rags handy.  
    A pair of channel-lock pliers, a hammer, a crescent wrench, and a phillips-head and flat-head screwdriver should see you through most everything else

2.  Turn off both water shutoffs and open faucet on sink to confirm water is off.  Disconnect the water lines at the shutoff.  Once you pull the sink out it will be much easier to disconnect them from the faucet.  
    Reinstall the same way.  Hook to the faucets on the new sink and then place the sink in the hole.  

3.  Remove the two screws to disconnect the drain from side of the disposal.  If you can, leave everything else in place.  
4.  There is a cam-lock ring that holds the disposal to the bottom of the sink.  The bottom ring should have a place to insert a screwdriver or some sort of tool to twist it to unlock.  
    Once the ring is free, you should be able to lift up and rotate the disposal and it will drop right out.  There are some tabs have to align (much like an AR Bolt) in order to drop it free.  
    You might have to tilt/rock the disposal to break free any sealant holding it to the bottom of the sink before you can lift and rotate it.  
    Careful, those things are fairly heavy and awkward, try not to drop it on your drain lines!  

5.  Unscrew the drainpipe directly under the right-hand sink.  You may have to remove/loosen the fitting at the bottom of the straight pipe in order to rotate the one at the top.  
    All of the other drainpipe/plumbing can remain hooked together and in place, you may need to give it a little support underneath with a box or something to keep them from sagging, tilting, or working loose.  
    There will be some water in the S-trap below the disposal where the pipes run into the wall.  If you can keep everything together, the water won't spill out and it will keep any sewer smell/gas from coming back up
    the drain line.  If it does, the water spill is minimal, so don't panic, that's what the bucket is for.  Worst case, remove everything and stuff a rag in the pipe to keep out the stinky.  

6.  Once all the water lines and drain lines are removed, remove the clips underneath around the perimeter of the sinks.  This should free them up to lift out.  
    Your sinks may have a bead of adhesive or caulking around the trim rings on top that you may have to pry or cut loose in order to lift them out.  A razor blade scraper or a putty knife should do the trick.  

7.  Don't stress.  As long as you've got the water shut off and a stub of drainpipe sticking out of the wall to hook into, everything else is just details that can be added to or removed when you have help.  

**  I'm sure a professional plumber will be along shortly to critique my instructions.  

Thanks so much for the replies.

Is it a dumb idea to leave the disposal attached and try to lift it out with the sink?

Why do you want to do that?

It's so easy to disconnect the disposal. They're heavy and you'll have to lift the sink pretty high to clear the counter with the disposal still on. Take the minute and drop the disposal first.

Guess I'm a tad worried about having enough vertical clearance to pull the disposal down. It's right above the drain pipe and only slightly off center, and I'd have about 1 1/2" to bring it down and then it would hit the pipe. If that's enough space to get it out I'll do it.

ETA: the picture is a little deceiving, the disposal is not off to the side of that pipe as much as it looks.

Please disregard that last post. Disposal is out.

Thanks again for all the tips.

Everything is disconnected from the sink and faucet.

Guess it's a cast iron sink. It wasn't clamped to the counter in any way.  It's pretty heavy, and I'd probably need an extra set of hands to get it out.

Kind of wondering now if I even want to replace the sink, or if I just want to clean up around it and just get a new faucet/sprayer and disposal.

Any opinions?

yeah those cast iron ones will hurt you

Clean up the cast iron. New stainless sinks are thin and cheap.

Man I'm inclined to agree, but the more I see of this sink the grungier I realize it is. There has always been a pretty bad rust ring around it between it and the counter, and the areas around the faucet holes are pretty corroded.

Let me ask you guys this: it's a 7" deep sink, and I just don't see any new cast iron sinks with that depth. The only 7" deep sink I see at HD or Lowe's is the cheap stainless. I want to use the same size because of the tight area I have vertically from the drain to drain pipe. As I mentioned earlier, there was only about 1 1/2" between the bottom of the disposal and that pipe.

Clean it up.  Wire brush off the big rust flakes, buy some POR (Paint Over Rust) to protect the bare case iron.  From the pics, the rest of the sink looks to be in good shape.  
Run a bead of silicone when you put it back to keep any water from getting underneath the faucet or the rim and you won't have to worry about any more rust rings.  
Unless you're looking for a deeper/bigger sink, I'd keep it.

Thanks. What do you mean by bare case iron?

You will need to unscrew the rusted hold down fasteners around the perimeter of the underside of the sink. And will need to  carefully cut any sealant loose under the sink flange.

Nice detailed instructions.  The part in blue

OK, had to clean up quite a bit along the edges where the sink was sitting, but all is good.

Gonna replace the old sink. The offset faucet has always been a pain. I want it in the centerline of the sink.

I appreciate the replies. Many of them were very helpful.

I have the new sink, and of course it's a little big for the hole. Looks like I have to shave off about 3/8" from the front and side.

If anyone can tell by looking at the above picture, what is the best way of going about this? I once had to make a bigger hole for my stovetop, but I used a reciprocating saw and made a jagged mess of it, chipping a lot of that thin material that covers the counter 

I think router would be best

It appears that the hole was cut in with a router originally. I would tape off just outside of where you need to enlarge the opening and trim it up with a circular saw or a router. taping off the laminate will help with chipping.

If you are careful enough you could probably use a reciprocating saw but make sure the saw is setting flush against the top of the countertop so it won't pull and rip at the laminate.

Putting the old sink back is still on the table 

Right now I'm scraping off the old sealant from the underside of it and hitting the rusty parts with a wire brush.

I have some rust inhibitor spray. Should I hit the underside of the cast iron sink once I get it cleaned up?

What sort or putty and/or sealant should I use to seal it back onto the countertop should I choose to reinstall it?

Thanks again.

Current pic of sink underside after a little bit of cleanup.

In case the ultimate goal isn't clear, when I started I had a nasty old garbage disposal, a faucet that leaked, a sprayer that didn't work, and a ring of funk and rust around my sink.

I am cool with keeping the old sink as long as it has been cleaned up and resealed to the countertop. I really just need a new garbage disposal, faucet, and sprayer.

Keep the old sink and install a new faucet/disposer.

That new piece of junk sink you bought isn't worth the effort to make it fit.

I agree. Got the old one cleaned up pretty nicely. Just need to find a set of hands again to get it back in.

Anyone want to chime in about the best way to seal it back onto the countertop?

Only if you get the cheapest ones.

Good ones are still pretty pricey and significantly thicker and stiffer.

Look at BLANCO.

They are NOT a 'bargain' but are worth the price.

Thinner ones will have a lot of vibration when the GD has a lot of water in it.

You can add framing (I have used wood, steel angle, and some rubber (or EPDM membrane strips) between the carbon steel and the sink bottom.

It helps with noise also.

I agree with the comment on the quality of new sinks, they do vary. Some food for thought, I've had cast sinks in the past, and while they are great, they can chip plates, cups, bowls, etc if they aren't placed in their gently. I personally would replace it with a quality new double basin. If it was me, I would put a bead of silicone around the perimeter of the cutout and set the sink in. Wiggle it a little to get it to "set" into the silicone, then wipe of the excess or even add a little more if it isn't completely sealed around the lip. Use plumbers putty under the new faucet and also under the drains. Youtube it if you need to get a better idea of what I mean. Any other questions, I'll do my best to help.

Went ahead and cleaned up the underside of the old sink with a wire brush. Sprayed some rust inhibitor on a rag and wiped it on the worst parts.

Put it back in last night and got a good silicone seal between it and the countertop.

Thanks again to everyone who posted.

You could have actually have that sink restored.  They do that the same way they do to old clawfoot tubs.