Place built-in bins for different waste streams
A lot of office kitchens/lunch rooms I've seen have only had one available bin (i.e only landfill). The designs of these kitchens did not account for having multiple waste streams - thus there was only one cupboard/cabinet or freestanding bin.
Some buildings awkwardly 'solved' this by having another free-standing bin for recycling, but very few of them had a dedicated bin for green (organic) waste. Even my current office building, arguably the newest one I've worked in (it was built this century), has two inbuilt bins for waste and recycling, and an awkward small 'container' on the bench for organic waste.
Thus, if you are designing the buildings and deciding where built-in bins go, ensuring you have appropriate bins for each waste stream would be a good step. Especially in the case of schools. If you are at all involved in the design of the playground layout, ensuring there are plenty of fixed bins for different waste streams: something like this:
Have a dedicated space for final garbage pickup
One office I worked in did not have a recycling bin anywhere. When I spoke to the secretary (who in turn raised the issue with the building management) - the response we got was they simply did not have the space to place a secondary dumpster to handle a separate recycling stream.
Thus, ensuring you give adequate space for garbage collection/pickup is essential. It's one thing to provide the bins in the buildings, but if maintenance/cleaning crews cannot dispose of them correctly then it's all gonna end up at the same point anyway.
Another factor on the design side to consider: make it clear what that space is for. If you leave it too open to interpretation - like just leaving extra space along a wall in the carpark - it might end up being the space they park the school bus, or the principal's private car spot. Clearly demarcate it as the place that the dumpsters should be: signs, dumpster facades, or sheds can work for this:
Incorporate a greenhouse/veggie patch
This one is more specific to the idea of designing a whole school, and might not apply in your case, but: design a space for a greenhouse (or even just a veggie patch), which could divert green waste from the bins into compost piles.
It would more than likely require buy-in from the owners/government education body, but the idea of schools hosting a vegetable patch for the kids to learn about plants and food has seen growing (ha!) support. If you're lucky, there might even be grants in your country/area to sweeten the deal!
One such site I found was Arcadia School Greenhouses who have some neat designs:
Design space for on-site composting
Similar to the one above - even without a big greenhouse on site - buildings with plenty of gardens/open space (such as school grounds) could benefit immensely from having composting systems built-in, with organic/green waste being diverted there for landscaping/gardeners to use.
This would of course depend heavily on the overall design of the buildings, and whether they have those open spaces that would make use of the compost in the gardens.
Place plenty of drinking fountains/refill stations
Especially in a school - reduce reliance on single-use plastic bottles from the canteen/lunch room by having plenty of bays of drinking fountains strategically placed around the school for the kids to use any time.
You could even use the kind that double as bottle filling stations, so that people can re-fill their water bottles as required, something like this one: