You DO have a very good climate.....for Central NY!

I was born and raised in central NY and we had a huge garden every single year. You can grow almost anything you want to, as long as your timing is appropriate. We grew everything by sowing directly into the soil.....sweet peas, many types of greens, beets, turnips, green beans, sweet corn, onions, tomatoes, zucchini, yellow squash, and much more. We also harvested asparagus every year, too.

My best advice to you is that you become very aquainted with your local Cornell Extension staff and take advantage of the wealth of local information that you will be able to obtain from them.

Visit the extension website and take advantage of the many fact sheets that Cornell has to offer on the subject of veggie gardening. Pay special attention to soil testing.

We had deep rich soil, but had to move about a million small rocks before planting every spring.

rhizo-1 is correct, you have loads of options! And the fact that you're already thinking in terms of succession planting is a good sign. In addition to the Cooperative Extension information, you can have a look at the chart in the link below for some basic veg plants and then look into which varieties of each would be best for your needs.

I'm technically zone 6, but unless it's something unique, I look for plants hardy through 5 as insurance to get the longest season - this winter's proving that method!

I always grow Juliet tomatoes - they're a grape tomato by definition, but for me they always produce very large "Roma"-size fruits and they are practically non-stoppable all the way through to frost. I have a relatively small kitchen garden so I buy transplants for tomatoes (I don't need enough plants to bother with seeding). For basil I'd start with a classic Italian Genovese - easily direct-sown - it makes a great companion plant for tomatoes (as do parsley and garlic, if you're thinking about an Italian-esque garden!) Just be sure to pinch out the growing tips along with whatever leaves you're harvesting, once it bolts (flowers) it's not so tasty and stops producing.

Zucchini is another easy to sow & grow veg. The only real consideration is that you might want to mulch under the plants once the flowers blossom just to keep the zukes from sitting on wet ground.

I could go on for quite awhile, but I think you have plenty to look into for now. If you have any more specific questions, ask away!

Here is a link that might be useful: A basic schedule for Zone 5

You sound like you are off to a good start.
- Plant things you like and know you will eat.
-Get a soil test done to see if your soil needs anything. Around here we need lime, but I don't know what if anything your soil will need. Adding organic matter to your soil is always good, and I usually add it in the autumn when I prepare the garden for the next spring. I use old manure or compost.
-I usually try a couple of different types of tomatoes to compare which I like and which produce best for me.
-Zucchini are prolific, so don't plant many plants.
-Herbs such as parsley or basil give good bang for your buck.
-Think about if you want to plant for late harvest and storage, like leeks, carrots, potatoes, onions, and winter squash. If you have a cold, but above freezing cellar or garage, you can eat veggies you have grown for much of the winter.
-Be sure you have a way to water your garden if there's a dry spell.
-Weeds are a reality in veggie gardening. I use mulch and/or a stirrup hoe and try to not let the weeds get too big.
-I love growing edible podded peas. They produce early and are great to eat raw or cooked.
- Beans are relatively easy to grow if you enjoy them. Like zucchini, peppers, basil, and tomatoes they need warm soil to grow.
-Peas, potatoes, many greens like lettuce, spinach and parsley are fine with cooler soil.
-ID bugs rather than just squashing them. Some will be predatory bugs that help eat the pests on your plants.

Enjoy the process! One of the nice things about a veggie garden is it gets redone every year, so if something doesn't work out you can try something different next year, and the things that work out well, you can repeat. And EVERYTHING is better than what is in the grocery store.

Greens are't just for spring! There are so many varieties for summer and fall. Many of these you won't find in your grocery store.

Thanks for all of the replies!

I read through pretty much everything on the Cornell Extension website, that was great but it has led me to have even more questions and concerns lol.

Should I be worried about the soil?

First, I live in the city and their website mentioned contaminants among other things in urban areas which sounded a bit scary. Can contaminants really get into the vegetables from the dirt?

Also I know that the soil is clay like and seems pretty thick. I do have some compost that I can either mix in or use on top but should I go beyond that? The city has a composting plan where you can buy 3-6 cubic yards of compost and mulch for $15 but you have to pick it up yourself apparently. I'm not sure how to get 6 cubic yards of compost to my house lol.

And lastly I was thinking of the pH of the soil, should I really do chemicals tests on it to see what it's like? The little garden will be near a walkway that is regularly salted in the winter time, could that have made the soil alkaline? For the record grass grew just fine in the area that I prepared last fall.

For the preparation I really didn't do much. I covered the grass in newspaper and made a small barrier with stones around it to hopefully kill the plants that we growing there, I have no idea if they will come back when I break the paper barrier and start planting though.

Thanks a bunch everyone! I'm going to search around for posts about soil now.

I think you can send/take a soil sample to Cornell to get testing done. They can probably test for lead if that seems likely to be a problem as well as pH, major nutrients, and organic matter. You can do some basic tests as outlined here:
http://organicgardening.about.com/od/soil/a/easysoiltests.htm
http://www.finegardening.com/how-to/articles/hows-your-soil-texture.aspx?id=81680

Adding organic matter to the soil is always good, particularly if it is clay (yours sounds like it has quite a bit of clay) or sand. Some options to get additional compost might include:
Can you borrow a truck?
Could you hire a landscaper to deliver city compost for a cost that seems reasonable?
Can you rent a truck (I think some of the big box stores rent trucks by the day/hour)?
Is there a soil/mulch/compost supplier that can deliver a quantity from their bulk stocks for a price that seems reasonable to you?
In future years, you may be able to pick up other folks leaves curbside to create more of your own compost.

How worried you should be about contaminents may depend on where you are planting. For instance I live on an old farm, so in general I don't worry, but I don't plant anything edible near the house or where other old buildings stood since lead was used in house paint for many years, so I just assume that the soil near the house has a relatively high lead content. Likewise, I wouldn't plant next to a busy road since lead was a gas additive, but if your land has no history of industry, I wouldn't worry beyond that.

Covering the grass is a great start since it will add organic matter as well. If you used enough layers of newspaper and it has been on long enough, most of the grass will have been killed. Grass will grow sideways into the garden under the stone border, so I would add a V shaped trench a few inches deep that gets refreshed a couple times over the course of the growing season to help keep the grass from growing into the veggie bed. Keeping mulch on the surface will help prevent weed seeds from sprouting and keep moisture levels more even.

There is a vegetable gardening forum and a soil forum and also I think a composting forum here on GW where you can also ask questions.