Hey, I use the same cubes as I'm unable to place orders (No credit card), and the rubiks speedcube is the only cube I have found in my country. I didn't work any lube into the inner mechanism. I also like vaseline for the rubiks brand cubes. They are fit very tightly together, and the vaseline helps break down the plastic enough that it turns better. Took me a good week to break it in, but once I did it was the best thing to happen to me. I went from 1.5 minutes average a solve to 50 seconds average in like 1 week of practice.
I popped one center piece, put a small amount of vaseline (Size of a small pea), and just performed lots of algs on it until it was worked through, I also wiped down the outside with a slightly soapy rag, then with a dry towel to get the grease off of the surfaces. It was very sticky, and got caught a lot on corners/locked up in the begining, but once the vaseline worked into it well and broke down the plastic a bit it's been a huge upgrade from the classic rubiks brand cube, which is all I had until now.
Also, after a week or so of use, you might want to take it all apart and clean it with a rag to get out most of the grease/dust and reapply the lube. When I reapplied I put the same amount on the center core, and spread it around, and then took the Qtip I used to spread it, and rubbed the edges of some pieces to make sure the glided nicely against each other.
Also, you can all bash me for using vaseline and bad cubes as much as you want. It's all I have, so I'll take it.