While I was there, I relied on the metro or bus, hoofing it on foot.
I can't vouch for what's cheap or the safest option – my approach is pretty universal wherever I go. Cheap is subjective, right? As for safety, well, I've always made it home "safe" from my travels so far. Some folks never even make it off the plane – a different spin on "safe."
Got a smartphone? If so, look into transit apps that might be handy. What do you use back home? Check if it works in different parts of the world. If it's a no-go outside your hometown, you know the drill, right? But hey, whatever app you choose, give it a spin at home first to get the hang of it. That way, you won't find yourself in the midst of BA, scrambling to figure it out in a rush.
I'm testing out Moovit, but there are other options. Google Maps is in my toolbox too, though not so much for transit – it's a bit tricky with buses and trams. Works like a charm for metros since the lines usually stand out. For any metro system, I download a PDF map to my phone for quick access. Also, consider staying near a metro station – it makes life easier. Best of luck!