You can grow your favourite vegetables in containers as all vegetables will do well in a container provided that container is big enough for the vegetable.
For example, if you try to grow even cherry tomatoes in a three inch diameter terra cot pot you are going to be disappointed; however, if you use a three gallon container or any container that is at least 24 inches across and 18 inches deep you will be able to enjoy tomatoes fresh from your own garden.
Add some basil which grows very well with tomatoes and you are all set.
You can grow cucumbers, green peppers and zucchini in the same size (24x18) containers. For both zucchini and cucumbers, you may want to add a trellis to the container to keep the plant growing upright.
For soil, I recommend that you purchase an organic potting soil, one that is designed fro growing vegetables, from you local plant nursery. If you are new to container gardening, it is worth the time to have a chat with the nursery manger and let her or him know what your plans are. They will be able to make suggestions for selecting vegetables that are ideal for your region.