We use a wooden frame (about 3'x3') with 14" hardware cloth stapled to the bottom to sift compost. This would work for your soil also.
You need to know that removing weeds and their roots is helpful, but not a complete solution. There are weed seeds in the soil, weed seeds blow in, weed seeds are brought in by birds complete with the accompanying manure, and weed seeds can lie dormant for years before the conditions are right for them to germinate.
Be careful of the soil you purchase to add to the beds. Some is mostly clay which is very difficult to work with, and some is just plain fill dirt that wouldn't grow a wart. How about adding compost instead? And hold off on the fertilizer until you get plants in and they are established. Too much fertilizer can burn plant roots, and the wrong fertilizer can encourage growth you don't want (high nitrogen=green growth, for example. Not something you necessarily want for a flowering plant).
I don't mean to rain on your parade, but judging from your posts, you need to find out what you are doing before you do it. Read up on soils, fertilizers, the plants you want to grow, and sun/water requirements. Gardening is an exercise in learning and patience.
