As mentioned by DA01, acid staining could be a good choice. There is just something to keep in mind with it though. Acid staining is somewhat transparent/translucent though. Is your patio free of permanent marks for example?
When my wife and I built our current come, we considered acid staining, but one of the cautions was that we needed to keep the concrete clean during the building processes. For example, some sub-contractors may like to make marks on the floor with a marker, chalk, etc. thinking it would just get covered up with tile or carpet. If it could not get cleaned up, it would show through the acid stain.
Now, considering that this is an outdoor patio, you most certainly don't have any construction marks, but you need to check and see if anything has marked the concrete since then.
If you do have marks or blemishes that would show through, and you still want to acid stain, then about your only option would be have someone put a new cream coat on the concrete. I'm not exactly sure what it is called, but my concrete sub had it done on a section of my driveway that was messed up cosmetically during the pour.
Finally, you could paint it. Just go and look for a concrete paint. Most can be tinted to any color you wish.
As an afterthought... You also need to decide what look you are looking for. Paint will give you a very consistent look, but acid stain will give a varied look--kind of mottled. Some areas will be a little lighter, some a little darker. It has to do with the way the acid reacts to the concrete. If you are not familiar with the look, just Google "acid stain concrete" and click on "Images".
Oh, by the way, the mottling is completely unpredictable, but that's part of the beauty!
Tile over green drywall is not waterproof; you will get water migrating through the group and into the drywall. The best thing to do is to tear down tiled walls, put up a new solid backing, and then put a waterproof membrane over the backing. Then tile goes on the backing.
Kerdi is a common backing, and that will be waterproof on top of a drywall backing, though I'd probably use a fiberglass-reinforced version.
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Another option would be to apply stone veneer to the faces and top of the steps, making it look like stone steps. I've considered doing that with mine.