Since this seems to occur at all times of the day and in different circumstances, I'd say she's partly frustrated by not having vocabulary, and she's also getting a huge payoff by having you bend over backwards to meet her whims. That's going to make your life a living hell - I know you think that's where you are now, but it will only get worse.
Reread Margie's response about putting a few pictures of healthy options on the refrigerator. Otherwise, put out the food that you are offering, and let her choose from those. If she doesn't like them, you say, "Okay, I guess you aren't hungry then. I'll put them away until you're ready." If she has a tantrum, she can "have some alone time in your room until you're ready to be calm." Separation from a parent makes a child realize that they have to get along, or be alone.
I don't understand what you mean about going out or using the basement. It's not about where you take a toddler, it's about what stimulation they have where they are. Rotate the toys, have a few in a bin or on a shelf where she can reach them herself. Are you saying you are offering her one toy to see if that's what interests her, and if it doesn't, you get another one out? It's better to have a few choices. You can get really simple organizer racks that also make clean-up easy. I put photos on the outside of each bin on a storage rack to show what went where: stuffed animals here, cars there, railroad track things elsewhere. You don't have to go out to be able to sit on the floor with a toddler doing a puzzle. Use toys in unique ways too - my son loved things with tracks (for cars, for trains) that could be arranged in different ways. Each day was an adventure. We had a table where he could set up and leave things up for continued play the next day (basement) with bins underneath for storage. We controlled how many bins could be open at one time so stuff wasn't everywhere.
The library is great - do encourage reading and let her choose some books to take home. Let her put them in the return bin herself so she gets the idea that things come into her life and go back to be shared with others. If she whines at any of these things, she goes home.
A play group is fine but I think we do our kids a huge disservice when we convince them that they must have something organized and structured. We've got a whole generation of kids who had this enrichment class and that team and this music group - and they are sadly lacking in innovativeness, critical thinking and problem solving. I can't imagine signing such a young child up for classes! So save your money and develop your own ability to create a stimulating environment - which doesn't mean you have to lead every activity! Most learning in children comes through free play - and they can do that themselves with a little bit of suggestion from the caregiver. There isn't a "right" way to play with toys unless they are a one-function item that doesn't require any thought or creativity. A dress-up trunk, a bunch of puppets, some "assembly" toys like Duplos or puzzles or things with tracks/roads are all great. So is driveway chalk - draw a "road" and let her push her kiddie car or just walk along it however she wants to. Blow bubbles and enjoy the shapes. Turn your dog-walking into a nature walk - take a bucket and let her pick up interesting stones (wash and paint them at home) or leaves (create a leaf collage) or pine cones (roll in peanut butter and bird seed, hang with a string outside as a feeder).
When you're at the library, get a book of toddler activities, things you can do at home that will spark your creativity. Show your daughter that she gets more out of doing then out of whining. It's not about you entertaining her to keep her happy all the time. It's about her being stimulated enough that she doesn't need just one solution to make her happy, so it's not up to you to keep guessing until you get the right one.