Cosmos are a doddle from seed and flower for months, so I'd definitely do them. They were flowering into early October last year.
Calendula, sunflowers, nasturtiums, french marigolds and poppies are also very easy to start from seed. The nasturtiums self-seed like crazy.
Sweet peas are a little more temperamental, but are so gorgeous it's worth a gamble. I pack a large plastic pot with as many empty toilet roll tubes as it will hold, fill the tubes with compost and plant a seed in each one. I water from below by standing it in a trough with water in. When it comes time to plant out, I dug a hole for each tube, which means the roots aren't disturbed and the root has been able to grow deeper than a small seed tray allows.
If you fancy veg, tomatoes, peas, french beans, runner beans, rocket and radishes are all very reliable germinators. With peas, I grow one trough for peas to eat fresh off the plant and one to cut as peashoots for salad. Cut and come again salads are very cost efficient and sprout very quickly which is nice and encouraging for a newbie gardener.