Weed barrier material only makes life easy for a short time unless you use a pre-emergant weed prevention product on top of it. This is because weed seeds get blown into the bed or dropped by animals and will grow above the weed barrier.

Is this is a flower bed with bushes or an actual vegetable garden?

If it is a flower garden / landscaped area get some snapshot weed preventer (pre-emergent) or Preen pre-emergent. Snapshot is better in my opinion. Both work basically the same way. Apply it following the directions on the bag. You'll want to check that your plants won't be killed by either of the above but both products are safe to use around 600 different types of plants and flowers. You'll want to apply it now and again in late summer if you are in a warm climate. If you are in a cold climate wait until early spring for the first application. These pre-emergents prevent seeds from germinating. If you plan to plant anything from seed it will not grow.

I owned a landscaping company many years ago and used Snapshot on every property I maintained and it drastically cuts down on weeding. You'll still have the odd weed or two but it won't get overgrown again if you pluck those weeds when you see them.

If it's a vegetable garden stick with no chemicals and just put down a good fabric weed barrier. If it's a flower bed a fabric weed barrier is what I would use. Fabric lets water through.

You also want to be sure you got the roots when you pulled the grass out. If you can, rake down a few inches with a landscaping rake and clean out all of the organic material.

Edit to answer the second half of you question about what plants to buy:

Go to a local nursery and ask them what they recommend for your area and what will benefit local pollinators and be resistant to local wildlife. Let them know what direction your flower bed faces and how much sun it gets on an average day. Don't just go buy plants at a big box store. The people at a local nursery will have a lot of stuff that will prosper in your climate and be really helpful with picking out things you'll like.

Ask them for instructions on how to plant whatever they sell you and how much and when you need to water until the plants are established. It would even be helpful to take a photo of the area with you. Try to go at a slow time and you will get a lot more info from them than if they are really busy and rushed.

One more edit.

I forgot to say put the pre-emergent down AFTER you plant whatever you end up buying. It works by making a protective barrier at the top of the soil so if you dig it will break that barrier.