What sort of plants can I plant on my balcony (I'm using shelving), and what can I get started on first? (April, May weather. ?

Where are you? You can google growing zones- they are based on the last date when frost is possible. Where I am, in the South, that date is April 15. I have put out some plants that will tolerate some cold- geraniums, begonias, annuals, and radishes, broccoli, and snow peas. I'll plant some tomatoes this week- although one year, it did freeze on April 14th and I lost a couple of plants. Here, it gets too hot very quickly for things like lettuce and snow peas, but peppers, cucumbers and tomatoes do well. Go to your local home improvement store or a nursery- if they have plants on sale, it should be safe to plant those plants;. You can grow just about anything in a pot. I wouldn't try corn, though!