Just wondering how long I can expect it to take my carrots, lettuce, cucumbers etc. to start growing.

Weather, amount of sunlight, water and heat all make differences--Then add in the variances of types of vegetables.
Radishes are usually the first to sprout-- the old timers would plant radishes with carrots because carrots are slow-- you'll pull and eat the radishes before the carrots come up. Beans are quick to sprout-- by the way, on the back of each seed packet the days to germination should be listed. AND the part of the country -- here in Oklahoma the ground is now warm and my veggies are cranking along-- had a wind storm today and my squash plants look kinda whacked from the window-- I'll check in the morning....
gardening is fun and tasty-- have fun and good luck

Read the back of the seed package.