Welcome to the world of gardening... if you'll persist through some dissapointments and count them as learning opportunities, you'll be well rewarded. Although I live in the western U.S., our gardening habits are not all that different from yours. Our winters are harsher but vegetables and flowers grow well.
I might suggest that, rather than the work involved in killing grass or removing sod, you might consider building a raised bed garden. This isn't difficult and the expense should be minimal. I'd suggest units of 10 feet long and 3 feet wide(or what ever suits your area). I made a new garden for my daughter (she of the decidedly brown thumb) this spring. I made it inthe shape of a capital "E", with the upright portion being 10 feet long and the three arms also ten feet long (by 3 feet wide). This provided a fairly large number of square feet with the ability to reach the crops from all sides and still be in a relatively small area. It's simple in that treated lumber (I used 2 X 8's) can simple be nailed into a box shape and then using treated 2 X 2's I made several stakes to drive into the soil at intervals to stabilize the structure. I then filled the boxes with top soil from the DIY store. Before pouring the bags of soil in, I place the weed matting over the grass inside of the boxes to inhibit grass growth. The soil, being 8 inches deep, warms rapidly in the spring and that can be enhanced by covering it with black plastic to give a jump on the growing season. The lawn can still be mowed around the shape one chooses.
Contd.