To flush any non-dual flush toilet, push the handle down all the way and release. Yes, on some you can get it to short-cycle, but it isn't guaranteed to work. You can call it a feature if you want, but is not designed or intended to be used that way.
Totos generally come from the factory with the water level preset properly, but you might check that. If the water level in the tank isn't proper, it doesn't have the head (or pressure) of the falling water to flush as designed. Then, make sure that the hose from the fill valve is pointing into the overflow tube. That is what refills the bowl. Both the bowl and the tank must be filled to their proper levels to work.
If the wax seal was not centered properly, part of it could be blocking the outlet of the toilet. If a wax seal with a plastic horn was used, on some flanges, it can get squeezed partially shut, clogging things (regular wax is the least expensive and best). Finally, if the drain lines are partially clogged, no toilet will work properly.
Last, if you are in the 1% or so that produces exceptionally large and/or hard stools for some medical condition, you may need a different toilet. The ones often suggested for that are those made by Caroma. If you use a lot of toilet paper, and it is something like Charmin, many toilets will have problems. Maybe a bidet seat would be a nice addition!
My mother has two Drakes in her home, and does not have problems.