In general I avoid using any sort of weed barrier liner, except on pathways lined with stones. At best weed barriers work for a season - weeds will eventually get into your soil and germinate.
when you are starting a new bed, the easiest, most ecologically-friendly way is to cut away any existing grass and then lay down a deep layer of good, soil. You can cover it in plastic if you like to keep weeds from getting established but avoid laying a liner under the soil, which isn't very effective and causes all sorts of problems wiuth drainage later.
It sounds like you have not mulched thoroughly enough, and/or your mulch is already contaminated with weed seeds. At a minimum I'll put down 3", and often as much as 6" around larger, fully established plants. Any less than 3" and weeds will be able to pop up from underneith. Mulch towards the end of the season and again in spring for maximum weed suppression and plant enhancement.
Don't use round-up. Just no.
If the area is large you can get a 'gentleman's hoe" which basically tears up the weeds. They'll keep springing back up but they are quick to knock down. Otherwise, hand-pulling weeds is just a fact of gardening, though if you need to do it more than once every 2 weeks something's not right (liklely hte amount/type of mulch, see above). I can hand weed a 2x10' raised bed in 5 minutes as long as i haven't let the weeds get too big and firmly established.