The only way one can amend any soil is by adding decomposed organic matter. I actually now have sandy soil and miss my clay. You can't use a rototiller on clay, you could but you'll be making things worse. Clay is made of tiny, tiny pieces of rock that are flat. These flat surfaces have electrostatic charges and these pieces are 'glued' to each other. By agitating the soil, mixing, you only increase these charges.
Think about how concrete is made. The ingredients are gravels, clay (Portland Cement), gypsum, lime and water. All thrown into a big drum and kept agitating until poured.
I've worked clay soils my whole life and it is a bit of work the first time you set up your garden. First, map out where your beds will be and walkways. You never want to walk on plant beds. For this answer, I'll be talking about vegetable beds.
My beds are 3' across or an arm's length from a walkway. If you have plenty of room, 3-4' wide beds bordered on both sides by 3' walkways is ideal. In my green house, I've reduced my walkways to 18" and my beds are 5' across with trenches all the way around so that water doesn't sit on walkways. Start digging by removing the soil from one end of the bed down 18" up to 2'. Dump this soil outside the bed temporarily. Then continue down your bed dumping the soil in the hole you've just dug. As you dig, have a wheelbarrow of decomposed organic compost and throw a bunch in on top of the soil you've just dug out. Don't worry about chunks. Chop a little but don't get carried away. You'll notice that clay is usually moist. I've tried drying it out and end up having to need a jack hammer.
Your 'fluffed' up soil will get a good 3-4' high, it will eventually settle down to 1 - 1 1/2 foot. I also throw in handfuls of 5-7-3 fertilizer that includes bacteria and mycorrhizae as I throw the compost on the virgin soil. The main thing is the bacteria and mycorrhizae which are essential for plant roots.
Rake and smooth this mound of soil and compost. Dig trenches or leave trenches 6"X6" all along the sides and ends, throwing the soil on top of your new bed. This will keep your beds draining and walkways dry. Don't try mixing any more, soil organisms will do that for you. Get a big piece of plywood lay it on top of the bed, get on top and jump up and down. This is to remove big air pockets. Your bed is now ready to plant.
I don't do 'rows' of anything. I have a salad bowl row where I put all my radishes seeds, all kinds of gourmet lettuce seed, carrot seed in a shaker (like the one you find at pizza joints that hold hot pepper flakes). I mix all the seeds together and very lightly shake seed on the top of my bed. Very, very thinly. I take a rake and standing on the opposite side of the bed I rake into the trench and sides to pull a thin blanket of soil on top of these seeds. It is a quick 'flip' that you'll get right away. Then I turn the rake (this one is the bow rake for raking rock) on it's head and lightly tap down on top of the soil and seed.
Otherwise I like to plant in triangles...I also make hills and squares for things like peas, pole beans, squash that need support.
After your plants get established, you can fertilize according to different plants'needs and always keep the soil covered with decomposed organic compost (DOC), not mulch that isn't decomposed. (Soil organisms will constantly be eating this stuff, taking back down into the soil profile, pooping it out, mixing this into your soil for you)! Make your nursery allow you to open bags to see, touch and smell the compost. Every batch is different. If you can see chunks of sticks, bark it isn't decomposed, don't buy it. Well, at least make sure that 3/4 or the material is dark, crumbly and smells like earth...not bark. The best way is to make your own compost so that you know what is in it...
In the fall as each of your beds are done producing, replant using a 'green compost' such as annual rye, buckwheat...they even have great mixes. They grow all fall and winter. This keeps weeds at bay and in the spring, all you have to do is turn the plants back into your bed. Clean your trenches and you are ready for the new season with great soil! You could make your walkways lawn, gravel, bark chips or leave as soil (clay is slippery when wet...). This even looks professional...don't forget to rotate your crops every year. Hope this helps.