I found a roll of "mat" that was available at a garden store, but I'm not sure if this would be very effective at keeping the weeds out. Water can get into it. It's made of some sort of shiny, synthetic fabric. It's very thin. I've taken pics of it.

Can someone please let me know if something like this, paired with gravel, would work?

Or do I have to buy expensive lawn fabric to get something thick enough to keep out the weeds? Please help. Thank you.

Most weeds and grasses can be smothered. Solarizing might work as a start. That involves wetting the grass and soil and covering it with clear or black plastic and sealing the edges from air infiltration. Do that for a month of very sunny days. Hot sunny days are better because they tend to help with the heating. Pull it away when you think the underlying weeds are dead. After that most of the plants underneath should be rotting away. As a further precaution you can continue to smother anything alive with sand and mulch.

Most people start with a product like RoundUp to renovate. Spray that once, water daily to sprout all the weed seeds, wait a week, repeat the spray to kill the new weeds, wait another week, and you should be good to go.