Two years ago I was a new beekeeper. Since then I have learned much, developed a bee allergy and been desensitized, and lost all my hives the first winter. This winter I brought through five out of five, three of which would not have made it without mite treatments. The fourth might have survived but got treated anyway, and the fifth I did not treat as it had almost no mites.
Here's my advice: Decide what kind of beekeeper you are. Are you the type, like me, who invests your heart and soul and a fair proportion of your limited financial resources into beekeeping? Or is beekeeping merely a side pursuit, like deciding to plant a few apple trees in the back yard, or getting a few chickens for fresh eggs?
If you fall in the latter group, you can go treatment-free. You will probably lose hives to mites, unless you can get bees from an established treatment-free apiary and continue their practices. Even then you might lose hives to mites, if other hives in your area are heavily infested. But it will be no huge loss, rather like if the deer munch those apple trees, and you will start again next year or decide beekeeping isn't for you.
If you fall in the first group, then I recommend doing all you can to keep your bees alive the first year or two. You can draw the line somewhere, e.g. no synthetic chemicals that leave residues in wax, but do not be afraid to treat for mites if you see that your bees have more than a few. Despite the loud proclamations of some advocates of "natural" beekeeping, using mite treatments does not make you impure, contribute to the ongoing plight of the honeybee, or generally make you a bad person. Yes, it is in the best interests of both bees and beekeepers to develop bees that can withstand mites without chemicals. But that is a long, challenging, and ongoing effort, and not one that you need to take on in your first year with bees. When you have mastered the basics and come out of winter with multiple hives, then by all means start experimenting with treatment-free approaches.
Approaches like drone trapping and powdered sugar dusting require frequent inspections and handling of the bees. This exacerbates the new beekeeper tendency to open the hive weekly or more often. Don't. It is easy to crush or drop the queen with novice hands, and frequent disturbance will decrease the already-low chances of getting honey your first year.
Formic acid, thymol, and oxalic acid are the mainstays of non-synthetic mite treatments. Commercial beekeepers use all three with good success. All can be hard on bees, but they do not leave harmful residues. Remember that bees are short-lived creatures, and the next generation will have no residual effect of the treatment but will be much better off for having lower mite loads. Count your mites in August, and treat if counts are high. Thymol and formic acid are good summer treatments. If they don't work (and they sometimes don't) or they don't work well, then follow up with oxalic in early December when the bees are broodless. Its effect is almost miraculous. If you had a heavy infestation you will see thousands of mites fall in a few days, and essentially zero thereafter.
Manage mites because you can see them. Live and let die is not a good approach if you are emotionally invested in your bees' survival. As for the other two chemicals commonly placed in hives, fumagilin and terramycin, I don't especially recommend them. Terramycin suppresses (but does not kill) the bacterium that causes American Foulbrood (AFB). AFB is nasty (infected hives should be burned, pressure-sterilized, or irradiated) but is not common. If you see AFB your first year you are either very unlucky, you purchased contaminated equipment, or your supplier has a problem. Continuous prophylactic antibiotic treatment breeds resistance and is generally a bad idea. Fumagilin used to be a reliable nosema treatment but is
significantly less effective against the new, now ubiquitous strain of the disease (N. ceranae). It is expensive and toxic to humans, and while it does degrade over time it is detectable in honey at low levels if used as recommended. If you really want to follow IPM, you can send a bee sample to a lab for nosema testing and treat if you have a high spore count.
You will probably not see AFB your first year, and you may or may not see nosema, but you will see mites in every hive. If winter survival is important to you, plan to do something about mites, and be open to using stronger, more effective non-synthetics (thymol, formic acid, oxalic acid) if softer methods are not working. These chemicals are a bit nasty in high concentrations, but they are all natural components of food (thymol in thyme and oregano, formic acid in fruits, oxalic acid in rhubarb and spinach), and any low concentrations persisting in honey will be entirely non-harmful.
Michael Bush will point out that essential oils (including thymol) and acids affect the microflora of the hive and bee gut, likely to the detriment of the bee. This is true, but mites vector viruses and suck hemolymph, to the very great detriment of the bee. Imagine one or two horseshoe crab-sized parasites attached to your body. They are a factor in most hive collapses, even if not the proximate cause.
Just the advice I would give to myself if I were starting fresh with bees...