In my honest opinion having been to many neighbouring countries, we don't have a good public transportation system here in Manila. LRT/MRT only has two lines that aren't very useful for many travelers, and buses are very difficult to use
For travelers, the best way to get around Manila is taxi or Grab (Grab is just like Uber. There is no Uber here in Manila anymore after Grab bought them out)
Taxis are cheap and plentiful. You can easily hail one on the street and get around town for a few USD. If you're at a hotel, they can help you hail one too. Taxis are all white and with a sign on top saying "Taxi". The biggest drawback with taxis is the potential for scam. Manila is notorious for having unscrupulous taxi drivers. You must ensure that the driver turns the meter on before you get on the taxi, because taxi drivers might agree to turn it on but tell you later it's "broken" or they "forgot". Payment is cash only, and make sure you have a lot of 20 and 50 peso bills because drivers will say they don't have enough change
I highly recommend using Grab to get around Manila. Grab's fare is similar to regular taxis, and there are a lot of cars on the platform that it's easy to get a car any time of the day. I recommend it because the fares are pre-determined and the entire process and transparent so there's very little chance for scams. Obviously you need to have mobile data or wifi in order to use the app, you can't just hail one on the street like you can with taxis. Like Uber, you can link a credit card to your Grab app so payment is cashless; but you can also pay by the driver by cash if you want
Other than taxi, in some areas of Manila you can also get around by tricycles (we call them "trikes"). These are three-wheel motor bike taxis that are more common in the rural areas and islands in Philippines, but they also exist in some areas of Manila (Intramuros for example). They are very cheap to use, most trips should cost you 30-70 pesos ($<1.5 USD). Unlike taxis, they are not metered, so make sure you agree on a fare with the driver beforehand
You may have heard of Jeepneys that many travelers talk about riding in Manila. They are like small shared bus that seat 14-21 people, that run in fixed routes that are not published anywhere. Unless you're going to a major landmark or transportation hub, I don't recommend you to take Jeepneys because it's very difficult to figure out what their routes are and how to get to where you want to go. If you really want to take them, ask some locals around you for guidance on which Jeepneys to take. Most Jeepney fares are under 10 pesos, and you can simply pay by passing the fare to the passenger in front of you, who will pass it on to the driver. Definitely don't take it if you have large baggage with you, and watch out for thieves!