Heh. Given my knowledge of physics, I'd be pretty dangerous in such a situation.
I'd teach some modern chemistry, physics, and metallurgy.
Smokeless powder is a no-brainer, of course. Improved cartridge-based, magazine-fed rifles would be handy, but that requires a massive production effort. If that is too difficult, some improvement might be obtained by simply using a cast Spitzer bullet with a rifled barrel instead of round ball ammunition.
More useful perhaps would be producing a smaller number of more useful devices.
A little primer on electricity and magnetism go a long way... Maxwell's equations, 30 years early? Simple crystal radio receivers are easy to produce in quantity. Transmitters require more work, but are still possible. Good communications are essential to modern tactics.
Now we get to the fun stuff... internal combustion engines and aircraft. It was difficult for the Wright brothers only because the principles were not well known. It's just a little basic thermodynamics and fluid dynamics. Build a couple small, light planes using cloth covered wood frames and small piston engines. Add a radio, and you have aerial reconnaissance. This doesn't even require the industrial base, as only a few aircraft are needed. (Did they have balloon reconnaissance at the time?)
The artillery of the day was primitive, IIRC in no small part due to primitive explosives technology and metallurgy. Better artillery would blunt an attack long before rifle range is reached. It isn't difficult to make RDX.
Proximity fuses are too difficult, but a simple contact fuse on a stand-off mount would work fine. Again, a Spitzer shell would improve range; did they have rifled artillery at the time?
Add man-portable tube-fired weapons. Mortars, rockets/RPGs, etc. would have been easy enough to make given improved explosives and metallurgy.
Finally: machine guns.
They're more difficult to produce compared to bolt-action rifles, but you don't need to make too many... a few hundred perhaps. A simple and basic design (probably recoil operated) is best. This requires a supply of decent quality cartridge ammunition; I'd put resources towards making machine guns and ammo rather than trying to re-equip every soldier with a bolt-action rifle.
