Dawnas,
You are going to have so much fun with your vegetable garden! It has to be one of the most rewarding experiences under the sun.
Start your tomato and pepper seeds indoors, probably now.
Garlic and shallots can be planted as bulbs in early spring or in the fall.
Cucumbers, carrots, lettuce and beans are direct planted into the ground.
Onions are put out as sets this time of year (they look like tiny green onions, and come in several varieties). Green onions are simply small onions, so I plant my onions twice as close in the row, and harvest every other one as a green onion. In the fall you can plant onion seed (Texas 1015 refers to the planting date, Oct. 15)but this ties up a lot of garden space for a long time.
Herbs in pots are great--just make sure that you get a variety that will fit in a pot! There are dills, basils and other herbs that get 'way to big!
As to storing the extra seeds, keep them in a cool, dark place. I put my extras in the refrigerator, but I usually have a second season garden in the fall, so I use up seed the year it is ordered.
Again, have fun with your garden! There will be some failures, but don't worry--it happens to us all and we learn from our mistakes better than from our successes (although successes are better for braggin'!)

Welcome to gardening in the southwest. I live in southern NM where the summer daytime temps are above 90 which make for limited or no fruit set of hybrid tomatoes, cukes, squash and the like. You live, I assume in the Phoenix area, where the summer daytime temps are 110 + and the night time temps seldom below 100 which is detrimental to any veg garden. As many zone 9/10 gardeners have learned, gardening begin is Sept and ends in June. Almost like you are live in southern South Am.

Other than that 'little twist' I agree with the above post with one addition. I store my seeds in a tightly sealed container in the fridge, keeps out the moist in the short term and prevents desiccation in the long term.

Happy and fruitful (pun intended) gardening.