If you don't want to get a weedkiller, you could start by harvesting the dandelions. Leaves before flowering and young leaves are used as salad greens and the "petals" are edible aswell. As are the roots, but they needs preparation. You can then cut everything down as low to ground level as possible, cover with weed suppressing membrane and then covering with mulch. Depending on how much mulch you have and how thick the membrane is that you get, you might have to spread whole newspaper with as little colour print as possilbe (numerous sheets at a time, not individual ones) over the area before applying the membrane and mulch. This will reduce a lot of weeds but dandelions are quite resilient and will probably simply weaken the more mature ones. If you can get around to using whatever method you use right away, at leave remove the flower heads before they get a chance to produce seeds. Every little helps! If you're up to it, turn the soil over first and remove as much of the roots as possible, this will slow down any regrowth. You may as well put whatever effort you can to minimise any recurrence right from the beginning.
