It's not possible to tell from your description what kind of citrus they are - but they do sound like citrus.
Can you post some photos?
Please don't destroy such wonderful old trees.
First you must deal with the suckers. Do you mean from the roots or from the trunk?
From the roots you need to dig down a few inches and cut them off.
If from the trunk, can you see a graft line? Everything below should be cut off level with the side of the trunk. If you can't see a graft line, are the leaves different on low down shoots? If so, again these are from the rootstock and should be removed. Otherwise, if there is clearly one main trunk choose a suitable height to produce a better shaped tree, and remove all lower limbs.
That's a start.

Unfortunately I am not able to upload pictures from my camera at this point in time -- I thought maybe I could scan a couple of leaves from the trees and upload them though. Will try to do that soon.

One thing I forgot to mention about the leaves is that they are very shiny, glossy. They don't look the same as my neighbor's citrus tree leaves.

On closer inspection of the trees, the suckers all appear to be coming from the roots. Someone must have attempted to completely remove one of the trees a while ago, as the original trunk is completely cut off nearly at ground level. A friend suggested to me yesterday that if I remove all the suckers except for one strong-looking one and focus on nurturing it, it could start to produce. Is that true?

The other tree has the original trunk, but I couldn't identify a graft line anywhere. The leaves from the branches at the top of the tree don't look different than those off the sucker branches.

I also have two lemon trees, with lots of thorns. (Are all lemon trees thorny?) One has a few small fruits in the interior of the tree. The leaves are all edged with brown and spotty-looking. Is this just from lack of water? I've watered it the last couple of mornings.

The other lemon tree looks half-dead, all the branches on the west half of the tree have scabby, scaly bark and no leaves. From articles I've read, it sounds like it could possibly be scorched? It is in the front yard with no protection from the hot afternoon sun. The leaves on this tree look similar to what I described on the tree, only not really green at all, yellow in the middle with brown edges and spotty all over.

What is the best way to start taking care of the lemon trees, regarding fertilizing, water, pruning, etc.?

Thanks for all your help!

Contact your local Agricultural Extension Office . They will have a citrus expert on staff to help you with all your questions .
You have a ton to learn but plenty of information available . Give both yourself and your trees time and patience .

This is the general link to your state : http://ag.arizona.edu/extension

Here is the link to the citrus page : http://ag.arizona.edu/crops/citrus/citrus.php

Thanks for the suggestion -- I'm sure it will be helpful!