#1 Start a compost pile if you have not already done so.
#2 Remove all weeds weed roots & traces of weeds & grass in the area that will be planted. Use a spade & double dig it you have to. This will loosen the clay soil for your carrots & other root vegetables.
#3 Dont wait to add organic matl. begin as soon as the weeds are gone.
Consider planting a nitrogen fixing cover crop that will make a good compost green, like peas.
These are only my suggestions, I have clay soil as well & the above worked very well for my needs. Part of the fun & satisfaction of it all is learning as you go & determining what works best for your needs. Hope this helps.

You may need to till organic matter in to your clay soil once, but if due care is used tilling most likely will not be necessary as long as you do nothing to cause that soil to become compacted.
Perlite is a mineral used in potting soils to promote drainage. It does not contribute anything in the nutrient area and to make much difference in the garden a very large amount will be needed. Far better is to spend you money, if necessary, on organic matter.
Keep in mind that Ma Nature has been at work making soil for far longer then we have and she does not spend money on exotic materials. Ma Nature takes the mineral portion of soil she finds anywhere and adds organic matter to that soil and then grows plants.

You may want to read into some of the work Ruth Stout does with her no till/mulch heavy approach. Gardening Without Work is one of her books and it is a must read and it has changed my perspective on traditional tilling in organic material and even fertilizing with conventional "organic" fertilizers

"Ma Nature takes the mineral portion of soil she finds anywhere and adds organic matter to that soil and then grows plants."

kimmsr -- love this simple way of putting it!

I'd say only till as deep as you have to and never deeper. Tilling(even the shallowest kind) is better than no-till in MOST situations, especially when you're also mixing in organic fertilizer and compost at the same time.

+1 what kimmsr said.