Hello, I have several tomatoes (mortgage lifter, Cherokee purple, Brandywine) growing in a raised bed with sun gold cherry, yellow pear cherry tomatoes, black cherry and super 100s. Most are heirloom though I don't know which are hybrids. Would I be able to save seeds for them even though they were planted next to each other (would cross pollination occur and give me different tomatoes?). I also bought Shikou, Vittorio and black beauty eggplants seedlings, crimson sweet watermelon and would I be able to save those seeds? Or lettuce and basil?
You can save seeds of all of these types of vegetables. Heirlooms, open-pollinated, and non-hybrids will come true to type from saved seeds; hybrids (labeled F1 on seed packets) will not be uniform in appearance or identical to the mother plant. Tomatoes are largely self-fertile, but some crossing may occur if different types are planted in close proximity. If you really want to maintain pure types (in vegetables that cross-pollinate), you would have to grow it in isolation from other varieties, to prevent insect pollination or wind-blown pollen. Some growers will cover the flowers with a bag and hand-pollinate the flowers, as a way to prevent cross-pollination. For most home vegetable growers, though -- and since tomatoes in particular are largely self-fertile -- it is sufficient to save the seed of heirloom types and you will get tomatoes that resemble the parent.
The following web page from the University of Minnesota Extension has a good explanation of saving seeds of different types: https://www.extension.umn.edu/garden/yard-garden/vegetables/saving-vegetable-seeds/