The Odyssey's a good choice. You could have gotten away with something smaller if you didn't have the requirement of also seating Grandma. And someone will probably still tell you to get a used Fit and squeeze Grandma in the back, or in the Thule roof rack or something, but the Odyssey will meet your needs a lot better, especially when you throw in a double stroller and two Pack-'N-Plays, etc.
New or used, that depends what you're comfortable with. My assessment is that they don't depreciate all that sharply in the beginning, so if you buy used, you're spending less, but you're often just buying something further along its lifecycle.
Interest rates on new aren't very high right now. Do not sell any taxable investments to buy this. If you don't want a huge monthly payment, take $10k of your $30k in cash and put it down; finance the rest.