I teach art and would like to make some strong 14"x 12" trays that the kids could easily carry and that can be easily washed. These are high school age kids and primarily sophomores to seniors, so 16 - 18 years old. Does anyone have suggestions?
There are lots of materials that can make a strong tray. The "easily washed" part is probably the biggest determinant.
I'll assume "washed" means with water rather than hydrocarbon solvents. How you wash the trays will make a big difference, also. If they need to be submerged and soaked, or hosed down, they need to be completely waterproof/submersible. If they can be sponged off with a damp rag, they only need to be water-resistant, and that opens up a lot more options. Another option is to use a cleanable or disposable liner so the tray doesn't need to be washed.
Is the goal to have trays or to make trays? If you just need trays, consider buying cheap, ready-made trays (cheaper and way faster than making them). If they need to be REALLY strong and indestructible, you can buy molded plastic food service trays that are roughly the size you need for a few dollars each, available from places like Amazon.
If they don't need to be quite so strong and indestructible, there are disposable food service trays made of styrofoam or compressed biofiber in the range of 50-60 cents each (here's an example slightly smaller than your target size, I didn't do an exhaustive search for your precise size: Bloomoon Heavy-Duty Disposable Food Serving Trays. Depending on the material and cost, you could use them a few times with minimal cleaning and discard them, or coat them to make them more water-resistant or harder so they last longer.
If the goal is to make them, here are some ideas.
Waterproof and washable
Corrugated plastic board. This stuff is very strong and waterproof. It's often used to make outdoor signs, and it's commonly used for trays and totes that contain a lot of weight. You may find precut sheets the size you need. There are standard size sheets roughly double your target size that could be cut in half. I think they're made from polypropylene, so they're difficult to glue. Construction with this material typically entails scoring and folding, and heat welding. Discarded signs can be a free source.
Cellular PVC house trim. This is sold as sheets, boards, and trim moldings at the large hardware chains. The brand-name stuff can be a little pricey; the chains often have their own in-house brand that's a little less expensive. If you know a builder or renovator, they often have large scrap pieces left over from jobs, and might donate it to the cause. You can cut it with woodworking tools and screw it together, or glue the pieces together with the PVC cement used for plumbing. It doesn't have the load-bearing capacity to use it for structural purposes in building construction, especially in freezing weather, but it is very strong and will stand up to anything you could throw at a tray.
If you have access to a local company that makes vinyl window mini-blinds, they often have scrap slats. Or if you know someone who replaced vinyl mini-blinds in their house, they may be happy to let you take the old blinds off their hands. These are much thinner than the cellular PVC sheets, so they are much more flexible. But they are still surprisingly strong if you secure the ends. They can be fastened side-by-side to create the bottom of the tray. You can make it even stronger, and keep the slats from separating, by adding another layer of slats running perpendicular. You can also put a little PVC cement between the layers of slats for additional strength and stability.
These materials will all require a little care and common sense for cleaning them without degrading the material, or use a liner to avoid the need to wash them.
Formica/melamine/vinyl-faced wood. This is available at big hardware stores. Sheets are typically used for backs of cabinets and bottoms of cabinets and drawers. Trim pieces are often wrapped molded wood composite. The laminate facing is easily cleaned. You mainly need to worry about water getting into joints, seams, edges, or an untreated backside. Use glue in any joints to seal the faces, and a little caulk or other sealer on seams and joints.
Wood/wood products. Marine plywood will stand up pretty well, especially if you add a good sealer. But you can also use regular plywood, OSB, Masonite, or other composite wood products, along with wood. Just do a good sealing job on the wood faces, and the same advice as the previous bullet for joints and seams.
Paper. You can make a strong, lightweight tray from several (two or three), glued layers of corrugated cardboard oriented perpendicular to each other. Cover that with at least one, preferably two, continuous layers of paper mache made with waterproof wood glue. Seal it well (two or three heavy coats, drying in between), with a waterproof varnish. This kind of sealing will hold up to temporary surface wetting, but it won't be submersible.
Cloth saturated with plaster or cement. These are very strong (a few layers of plaster cloth are used for casts for broken bones). A single cloth towel saturated with cement and laid on a form to harden will make a strong tray. Even though cement is heavy stuff, you don't need a huge amount of it in a tray. After the cloth is on the form, vibrate the form, then top off the surface with more cement to make it smooth and stronger before the cement has started to cure. You can form handles with the cloth when you put it down. Cement can be a little abrasive, so you may want to sand the surface after it's dry.
Plaster will work better with several thinner layers, each applied before the previous layer has set up.
With either, saturate the surface with sealer after it is fully cured.
The question doesn't provide details on the intended usage, so you will need to assess which of these approaches will provide appropriate strength and washability for your application.
One can find HDPE (or maybe it's LDPE) cutting boards in various sizes and thicknesses. One can also purchase sheet polyethylene in various thicknesses. Using this material in a manner similar to wood, a simple tray can be constructed with coarse threaded screws or wood screws.
This type of plastic rejects most adhesives and is easily cleaned of paint, glue and materials that may adhere to other types of surfaces.
As noted in the comments, more information regarding your capabilities and limitations would better define the answer.