Just what to do depends on what you do have. Annual "weeds" should be no real problem since they die every year, but perennial unwanted plants can be. However, if you deny any plant access to the sunlight they need to grow (covering them with something to keep that sunlight away) they will, eventually, die and contribute the nutrients they removed from your soil back into that soil. Yanking plants out and throwing them out just removes valuable nutrients from your soil that yuou then need to replace.

Thanks, kimmsr! I'm pretty sure that they are perennial, since I live in a tropical area.

If I use the mulch/newspaper method, how will this affect the roots? Will they be easier to pull out after the weeds have died? Or will I still be better off using a rototiller to dig them up?

Covering a plant with something to keep that plant from getting the sunlight it needs to grow will cause that plant to die and as the food reserves stored in the roots are used up attempting to grow with no more food being manufactures by photosynthesis those roots will also die, and provide additional nutrients to the Soil Food Web, which will feed the plants you want to grow. There is no real need to remove most "weed" roots unless what you have is something like Quack Grass which seems to live almost forever.