Check valve goes like this: 2L juice bottle - tubing -check valve-syringe - tubing to diffuser inside tank.
The check valve should go in the direction that allows CO2 gas from bottle to enter the aquarium. You can blow through it to make sure it's facing the correct way.
The tubing will fit directly on the syringe output and also on the check valve, so no glue is needed there. Only glue is needed where the check valve and syringe connect.
(picture from a member on another forum).
Either tubing will work, silcone will get brittle over time, normal vinyl tubing will get harder over time. Both will need replacing once you see that it does or they system starts to leak CO2. This time can be a period of 3+ months before needing service.
-John N.