I have used a lot of finishes over the years but have recently settled on a combination of dye, shellac and oil that I think is easy, safe and produces good results.
I don't know what your safety concerns are, but I think about the toxicity or flammability of the materials, and the amount of fumes produced and for how long. I have started using clothing dye as a stain (RIT Dye, which one can acquire at grocery, variety or fabric/sewing/hobby stores) for a stain, though I still occasionally use oil-based stains since I still have a lot on hand. The RIT Dye has no odor that I can detect, is easy to apply and very forgiving. If the result isn't quite right, it can be removed or lightened with light sanding, darkened with additional applications, dries quickly and is inexpensive. I believe this dye is non-toxic, but recommend reading the label. Wear gloves, because it dies everything, including skin.
I usually follow the stain with a few layers of shellac. Shellac is also easy to use and very forgiving. It dries very quickly. Quick drying allows multiple coats in a short time; sand lightly between coats with fine sandpaper or steel wool. Shellac contains denatured alcohol, so it has that sort of odor, and produces those sorts of fumes. They dissipate quickly, though. The liquid material is flammable, as are fumes in the right concentration, though I've used it indoors many times with no problems. If you want a quick, easy finish, shellac fits the bill. If you're not using stain, and don't want to impart a yellowish color, look for clear shellac.
After shellac, I apply multiple layers of my current favorite oil finish, Birchwood-Casey Tru-Oil. Tru-Oil is mostly boiled linseed oil, with some other ingredients. Boiled linseed oil is the main component of oil-based varnishes, so Tru-Oil is similar to Danish oil and the like. Tru-Oil is easy to apply, one can use a rag, or a gloved hand. Allow time to dry between coats, sand lightly using steel wool. I usually apply three or more coats; each successive coat takes less material. Tru-Oil is flammable as are its fumes; again, I've had no problems applying indoors.
There are other options, of course, but I've tried a lot of things and settled on this process for wood that will be stained or left natural. You can find about plenty of information by googling; lots of good videos on youtube regarding finishing wood. When looking at finishes, keep in mind that water-based does not automatically mean non-toxic, nor does it automatically mean low-VOC. I have used some water-based varnishes, and was surprised at the amount of fumes. They were more bother than shellac and Tru-Oil and the results were not as good, and I found the fumes more objectionable than shellac or oil.
As always, the usual disclaimers apply - your mileage may vary, objects in mirror are closer than they appear, past performance is no guarantee of future returns - and be sure to follow label instructions and assure adequate ventilation.