What's a person to do? Well, first of all, know thine enemy.
Rodents can get in via very small spaces around pipes and new construction doesn't necessarily exempt you from that infiltration. New construction, depending on the quality of workmanship, might actually be worse than old construction. Obviously, prevention is best, but rodents are very persistent and will seek out, and find, ways to get into your house or shed.
What you'll notice with rodents is their urine and/or excrement, damage to containers of food, missing pet food (cat food was a favorite of the one that chewed his/her way in under the sink near our dishwasher, which had rotted out the layer of flooring underneath it due to a bad leak), and spots that look like something's rubbed against them repeatedly. You might also see gnawing around entrance areas if the rodents are taking advantage of existing holes.
Rodents can get through very small openings, and you should ensure that none are be larger than 1/4" inch, under doors, around pipes, etc. If you find possible entry holes, or holes that could be used by rodents, stuff them with steel wool (steel wool scrubbing pads will work if you can't find regular steel wool] and staple hardware cloth around the vulnerable areas. You can also use foam made to fill voids around doors, pipes and other openings, but you should use steel wool as well.
Rat and mouse droppings look different; mice leave little 1/8" turds, pointed at both ends, while rats' tend to be about four times as long and slightly curved.
Control and elimination requires diligence in determining how and eliminating the ways the beasties are getting in, plus removing sources of food such as pet dishes and food that is only in plastic bags, such as English muffins on the counter or an unopened bag of tortillas. Although they can chew through thicker plastic, like Tupperware or butterware, they tend to go for the easiest access first.
You may need to clean out all your cupboards and scrub down your counters, depending on the amount of rodent activity you discover. Until you get the issue in hand, it's a good idea to be extra vigilant with cleaning surfaces and handling foodstuffs.
When you put out traps, the best place is along a path that is traveled by rodents, which usually like to keep to the perimeter of a room, along a wall. While there are sticky pads you can put down, the more conventional traps are better because they won't lose their stickiness due to dust collecting on the surface, but put out both so that you have multiple methods working for you. Put mechanical traps perpendicular to walls along paths, and bait them with peanut butter, wool yarn scraps, cat food, etc.
Rodents supposedly don't like the ultrasonic repellers you can purchase, but we didn't have any luck with them when a couple got into the garage. Warfarin works (read Wikipedia's article on the stuff); note that it works because rats will return to a food source that they don't get sick from, and it takes several days to build up enough in their systems to actually make them sick, and at that point, it's too late.
Rodents will also go after nesting materials, and that can be things like terrycloth towels, yarn, dust bunnies (that's the only explanation I can come up with for the one in the middle of the kitchen floor] and...dryer lint. That came as a shock; we found evidence of rodentia in the laundry room, and moved the washer/dryer stacker, and...ew. Yuck. Blech! Rotten little critter had been hiding under the stacker and had left his/her calling cards...many of them.
Obviously, as small as the openings are that they can get through, you'll need to be thorough and vigilant in eliminating hiding places as well as food and nesting material sources.
Bugs that infest flour and other goods are easily dealt with by 1) vacuum packaging the material with two medium sized bay leaves in the package, and 2) freezing the package.
It's anecdotal, I know, but I've never had a problem with flour or whatever if I put bay leaves in the container. If I vacuum package something like flour, I include bay leaves, and also freeze the package. I figure between the two techniques, what I store should be fairly bug free. Storing flour in containers with bay leaves also helps. It also helps to store small amounts, and vacuum pack small amounts, rather than store or vacuum package large quantities.
If you store things in vacuumed canning jars, bay leaves are good there too. I'd suggest using an oxygen absorber in a canning jar if you are 'canning' something like penne pasta or macaroni that has spaces between or in the individual items. While I've 'canned' things like sugar and rice, I'm leery of relying solely on the vacuum to pull out the O2 for items that aren't as dense.
A widely used technique for storing grains and beans and the like is to put the item in a mylar bag, seal it (with or without oxygen absorbers, most likely with] and then put the bag in a five gallon sealable container. The only problem I see with this is that it's hard to use up that much quickly enough to avoid contamination by pantry pests or having the stuff get rancid or absorb odors. I vacuum pack according to the size of the containers I have in my cupboards, and that's about 2 1/2 - 3 pounds of an item in a vacuum-packaged bag. You'll need to decide what works best for you; it's not a problem to use multiple mylar bags of multiple items in a five gallon bucket, but it will necessitate good record keeping of what is in inventory.