Beirut does not have a metro or tram system, but you can get around reasonably well once you understand the options.
Shared taxis (service) are the cheapest way to move around. Flag down any car with red plates, say "service," and the driver will take you along their route for a flat fare of around 200,000 LBP ($2). You share with other passengers heading the same direction. It takes some getting used to, but it is efficient.
Private taxis are everywhere. Say "taxi" instead of "service" and agree on a fare before getting in. A ride across central Beirut should cost $5-10. Uber and Careem both operate in Beirut and are a good option if you prefer fixed pricing and app-based booking.
Buses run along 11 routes in greater Beirut, with fares starting at 70,000 LBP. The system is still relatively new and growing. Routes cover the main axes of the city but do not yet reach everywhere.
Walking is feasible in specific neighbourhoods -- Gemmayzeh, Mar Mikhael, Hamra, downtown -- but Beirut is not a particularly walkable city overall. Pavements are uneven, drivers do not yield to pedestrians, and distances between neighbourhoods can be significant.
Driving in Beirut is chaotic. Traffic is heavy, parking is scarce, and road rules are loosely observed. If you rent a car, you will need patience and a certain tolerance for creative lane management.