Unless you’re hitting some remarkable exhibit you can show up for most museums and be just fine. As for dining alone, sure, you may receive some attention, but if you’re civil and clear with your “no,” would-be “escorts” will leave you alone. The trick is never seeming flustered or angry. And don’t be surprised if you’re the only woman dining alone. Just enjoy the meal. As for the day trip. choose Siena because it has immense flavor and the whole of the city is a marvel. Pisa offers mostly the tower and the remarkable monuments around it. After that, it’s mundane. Siena is layered. You can still get lost in it. And the main piazza (Piazza del Campo) is a wonder of the world.