Unless you are really bothered by the sight IN THE SHORT TERM, you might just wait a while and see how that portion of the concrete weathers, now that it will be exposed to the elements in the same way as the portions surrounding it HAVE BEEN.
Bleach the concrete. Non-clumping kitty litter (the clay type) will also absorb anything water or oil based out of the concrete.Dry portland cement worked into the surface will also absorb any oils if they are involved in the staining.
Using an oxygen bleach may work where chlorine does not, or vise versa. The last thing I would do is try to make paint stick to and cover a stain.
A pressure washer may be required to get it clean too.
I've used Benjamin Moore two part industrial epoxy to do several garage floors. It's great stuff, lasts covered for over 10 years depending on traffic. Uncovered, it fades a bit over time, but still looks good. Comes in almost any color, but unless you have a unusual decorator sense, the light greys as close to natural concrete are probably best.
If the concrete is more than six months old and the footing was properly laid, there shouldn't be an issue with moisture. If the concrete is already spalling (chipping off or crumbly), epoxy won't preserve it.
It's not cheap - the good stuff is sold by the gallon and you need a gallon of each part.
Before you go that route, you might consider cleaning the concrete with muratic (swimming pool) acid. Nasty stuff, but it's also recommened as prep for the epoxy, so if it doesn't work you haven't wasted the effort.
I'd try one of the products for iron stains and similar, available in the plumbing dept at HD, etc. If it's iron based, they make it disappear in seconds. And whatever it is, good chance they would work.
The lovely Miseries Snuffy has the same problem with the front porch - and hence my problem... Her idea is to lay bricks. I told her good idea, go ahead!
My vote would be to not paint the concrete steps. Once you paint them, you cannot un-do that option. Once painted, they can become more slippery, they will have to be repainted every so often, and if they are outside I think they will begin to peel and flake sooner or later anyway due to the weather, water, freezing, etc.
You may want to try the pressure washer idea that others suggested. I recently pressure washed a house including the concrete block foundation on the outside. The pressure washer worked well on the concrete block (and everywhere else). I had never used a pressure washer before. I rented an electric pressure washer from Home Depot and it was easy to use. I only used plain water in it and no detergents or bleach etc. and that worked well. But, you may want to try adding bleach or detergent for your use. Others here may know if adding bleach or detergent would be a good idea or not.
If you do get this resolved, let us know what worked and what did not work.
My experience with using a pressure washer on concrete is that they are great at removing dirt, grime, mold, etc, but not effective at removing stains. I've seen what she has, ie stains from flower pots and I doubt a PW is going to remove them, because they have leached into the concrete. I'd try one of the rust/stain remover products made for toilets, sinks, etc.
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