Is it better to take taxis or the subway?

Always take the subway. Traffic in Manhattan is notoriously gridlocked, and a taxi will cost you triple the price and double the time.

Almost nowhere. Nearly every shop, restaurant, and transit turnstile accepts contactless payment. Keep $20 for emergencies, but you won't need much more.

How to plan your first 48 hours in Manhattan

Stick to one geographic area per day to minimize subway time and maximize sightseeing. Prioritize walking and use the MTA OMNY contactless system for all transit needs.

Master the subway grid. Don't buy a MetroCard. Just tap your credit card or phone at the turnstile using OMNY. Trains running 'Uptown' go toward higher street numbers (North); 'Downtown' go toward lower street numbers (South).
Book one 'major' ticket in advance. Pick one high-demand attraction like the Top of the Rock or the 9/11 Museum and book a timed-entry slot at least 2 weeks before you arrive to avoid waiting in lines.
Define your radius. Group your activities by neighborhood. Plan Day 1 for Midtown and the West Side, and Day 2 for Lower Manhattan and the East Village to avoid crisscrossing the island.

Is it better to take taxis or the subway?: Always take the subway. Traffic in Manhattan is notoriously gridlocked, and a taxi will cost you triple the price and double the time.
Do I need cash?: Almost nowhere. Nearly every shop, restaurant, and transit turnstile accepts contactless payment. Keep $20 for emergencies, but you won't need much more.