Sounds like you need to just clear that bindweed first of all! You need to use a weedkiller that contains GLYPHOSATE. This type of weedkiller will kill anything with green leaves - it is absorbed by the leaves and taken down into the plants roots, then kills the plant from the root up, so the plant won't grow again. The best thing about it is that once it is in the soil it decomposes and becomes harmless, so you can plant into the soil immediately - the ground is not poisoned. So, you need some glyphosate (for example, Round-Up weedkiller). Spray it (or use watering can) all over the bindweed. Then WAIT - you must give it time to be taken down into the roots, this can take up to four weeks or so, don't be impatient! Once the bindweed is clearly dying, you can dig / pull it up (carefully, try not to break off too many bits of root). To be on the safe side, wait another few weeks to see if any regrows - if it does then repeat the treatment. This should leave you with something like bare soil - that's good, because you don't want a lawn if you are going to try to create wildflower meadow. You can plant then; although depending on what you want to put in you might be better waiting til spring.
And yes, a lawn weedkiller will kill your wildflowers - it is designed to get rid of anything that isn't grass.
Others on this board will give you better advise than me, I'm no expert! Enjoy your garden and take your time - don't let the weeds get you down, I promise you will beat them in the end! Let us know how you get on.