It does take longer for weeds to appear than it used to. I'm not sure when it changed, but it can take a week before weeds show up. As oshizo said, you can raise your Sim's gardening skill quickly by planting fast-growing plants and tending those (spinach, basil, parsley, sage, snapdragon, carrot, potato, lily, daisy, crysanthemum). You also gain gardening skill by harvesting plants in the wild and by interacting with the cowplant.
I like to plant 40-50 fast growers. Plant, water, evolve when ready. Once you get to level 5, the fastest way to gain the skill is by taking cuttings and grafting. Also, being focused will increase the rate of gain.