On that first one, those roots are aerial roots. You do not need them to be in the soil, and most people just let them be. Often, those roots are a sign that the plant is being underwatered, and the plant is seeking a new source.

That next one will be a bit trickier. That brown near the bases looks so much like rot, that I will go ahead and say it is. You will need to cut out all the rot, even if that means cutting the healthy parts from the roots. Use a nice sterile knife to do the cutting. Cut back rot to healthy, untainted tissue (you will be able to tell by the color). I would cut some of the healthy tips off in about 4 inch segments, letting them sit somewhere warm and dry (out of soil) for several days to callous over, and place them in dry soil (tops pointing straight up). In a couple months, the cut segments will produce roots, and you can start over with a healthy plant. I am not sure you can save the mother plant, as it looks too far gone, but I would try to cut off some segments and root them.

A couple general things:

Both plants look like they need more sun, especially the lower one. Acclimate them to the sun slowly, so they won't sunburn.

In my opinion, they do not yet need larger pots. Succulents and cacti like to be a bit crowded.

If you want to do a bit of reading, you can do this:

http://forums2.gardenweb.com/discussions/2842847/container-soils-water-movement-and-retention-xxii?n=531

It contains a lot of info about proper container soils. You can search in the forum search bar for recipes, or ask questions yourself.

As to ID's, I don't know much, but the second one might be some sort of opuntia.

Good luck!

~palmsandsnow

The first plant is likely a dragon fruit - pitaya, hylocereus. She may have grown it from seeds. This plant grows pretty large and best on a trellis/support, they are epiphytes - you can google for photos. The aerial roots could root themselves; in wild, they do that and search for more nutrients. At home, I believe they can be cut off. They bloom overnight, are could bear fruit - flowers are pollinated by moth and bats (?). They need quite long warm season, so fruiting also depends on where you are. Plant needs to be mature to flower.

I almost think the dragon fruit was the root stock part of a moon cactus graft - could that be a dried up gymno on top?

Leslie-

I think you might be right! Looks about right for a moon cactus.

Good eye Leslie, you are probably right...Hylocereus is one of the rootstocks used for grafting Moon cacti...

Should I leave the moon cactus part on top even though its dried up? Does that matter?

Your grandma would be so happy to see you working on her plants. They do need quite a bit of help! Be sure to get pots with holes in the bottom.

The second one is Austrocylindropuntia subulata.