Re the flowerbeds, yes, you need to dig them over and extract all weed roots, preferably without leaving any behind. The guy at Home Depot's advice was incorrect though - if you want to use Round Up, that works 'through the green', which means it needs to be applied to strongly growing green growth when the soil is damp. Applying it to roots or bare soil does nothing other than cause a bigger residue of Roundup in the soil.
He's also wrong about how to use weed membrane - if you want to use it, lay it after you've dug the beds over and let them settle, having left enough time for any regrowth of weeds to show itself so you can dig that out too. Once you've laid the membrane, you will need to make cuts or slits in it and plant through those, so the roots of your plants are in the soil of the bed beneath, and the topgrowth above the membrane. That can be a pain if you like to use temporary plants for summer - every time you want to replant, the membrane will be in the way. After planting, cover the membrane with some type of mulch, such as bark chips or similar, or an inorganic mulch such as the stones you mention. Alternatively, don't use membrane at all, just rely on hoeing out any weeds as they appear on a regular basis around any plants you've put in. If you choose not to use membrane, then it would be better for the soil and your plants to use an organic mulch rather than stone. Membrane isn't a perfect solution - some weeds are difficult to get rid of and keep regrowing under the membrane, and other weed seeds germinate in the mulch layer and grow there instead, so sometimes its easier to just keep hoeing, so think about which you'd prefer to do.
As for the grass problem, there are, I believe, herbicides which kill crab grass - I'm not sure if Round Up does, it should say on the pack. Once you've killed it, you need to read the directions on the products you've used - there will likely be a gap or length of time from use of the herbicide and when you're able to reseed or lay sod, so wait the required time. When it comes to reseeding, you will need to create a fine tilth on the bare soil (that means a fine, crumb like structure for the top inch or two), then sow into that. This fine tilth is also necessary for sods to root into easily, but take care that the level you lay the sods at is one that matches up with the rest of the grass.
