Don't cheap out on the oils and get an artist grade. You might even want to invest in artist grade thinners for cleanup, although your hardware store mineral spirits will work. Cheap makeup sponges and cotton buds are your friends for cleanup.
I never worked with oil paints and thinners before, so I don't know what brands I can find in my country and city. I would appreciate it if you could mention some oil paint brands that are worth buying, and I will check if I can find them locally or I would have to order online.
As for the makeup sponges and cotton buds, I've seen people mentioning that regular cotton buds are a no-no, because their material frays and they can leave fibres stuck in the oil paint during cleanup. Can I substitute cheap makeup sponges with cheap makeup applicators?
Left field: make your base colour(s) lighter than you want them to end up, then cover all over with Citadel shades (aka washes). Nuln Oil black, Agrax Earthshade brown, and Biel-Tan green are terrific washes. They'll settle into the panel lines and darken them reliably, you just have to avoid pooling by using a dry brush to sponge pooling off. They're also not too thick, so you can use two coats if you want to darken things up.
The trick is to recognise the wash will darken the mini, so plan accordingly.
In certain cases (this being one of them), I want my minis to be the exact base color I chose (I love Vallejo's Model Color Olive Grey and Russian Uniform, perfect for military paint jobs and camo), and I would not be able to emulate that by washing a lighter color.
You can still use the contrast paints for recess shading by painting it into the recesses instead of covering the entire model and then going back and repainting the raised panels.
I wanted to avoid pin washing as I thought that it would take more time than I was willing to spend on that step, and I would probably miss recesses and panel lines somewhere.
Because I'm That Guy, I prime in black and drybrush the whole model.
Drybrushing the base color on is something I have thought of, but ultimately didn't go with it, as I wanted a nice even coverage with the base color I chose for these mechs, and due to my hands shaking, I tend to make mistakes, which would be significantly harder to fix on a drybryshed surface.
When I saw that the current procedure was taking me so long, I thought that I would be better off by applying the base color directly on the black primer, paying attention to recesses and panel lines, as I basically already did that to fix the darkened base color by the wash.