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!
As long as the steps are not attached to anything and the patio has a good mortar base and a packed gravel underbed then it should support the steps.
During the winter ground moves up and down due to freezing/thawing, so the steps cannot be attached to anything, because the whole patio will be moving up and down during the winter.
If the patio is old and has been been through several winters and shows no signs of cracking, then it is probably good to support stone steps. If it has cracks already, the steps will likely cause more cracking and settling. If it is a new patio, then it is hard to tell, because it depends on how well the patio was made and the soil.
Therefore, if it is a new patio, it would be better to use a separate foundation for the steps. If the steps are disconnected from the house, then you can use a gravel footing. Make sure it is drained with a pipe going to the storm runoff. If it is attached to the house, then it should have pillar footings that go below the frost line (about 4-5 feet deep).
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Tightening the screws is really important you might want to replace some with slightly longer as the holes over time will not hold well (if they hold you are golden). sanding with a coarse grit to get the paint off is a good idea to help reduce the high spots also the heavy weathering After sanding you will wand to clean the deck to remove the dust from the grain and screw heads. I have used a product that home depot sells called Restore. this looks nice but I think it comes in brown and Tan. Restore will help hide the heavy weathering and you really don't need to finish sand only rough sand or pressure wash to get rid of the loose paint. I have refinished horribly weathered decks with Restore that came out looking really nice. The first one was 6-8 years ago and it still looks close to the way it did when I put it down. No matter what you use the boards are "dry" other than minor water penetration and you should have no problems with cupping or warping. Exposed boards really warp and or cup much faster than sealed boards.