The right frame
Metro Manila is not one place. It is a collection of cities with different characters, and conditions on one street can be quite different from those a few blocks away. That is why blanket statements about whether the city is safe are not very useful, and why the single most valuable source of information you have is the people at your accommodation.
Ask them two specific questions on arrival: which directions are comfortable to walk in the evening, and where the nearest reliable transport pickup point is. Those two answers will shape your week more than any general advice.
Transport is where most problems start
The recurring issue for visitors is not violence, it is being overcharged or taken somewhere other than where they asked. The systems that exist to prevent that are worth using deliberately.
At the airport, transport operates through an accreditation system in which authorised vehicles pick up passengers from designated booths at transparent rates, and the operator has publicly strengthened those measures after an overcharging incident. Staff hand you a dispatch slip when you take an accredited taxi, and the instruction is to keep the slip rather than pass it to the driver, since it records which vehicle you boarded.
In the city, booking through an app gives you the same benefit in a different form: a recorded trip, a named driver, a plate number and a price. Check the plate before you get in rather than after.
Money habits that reduce risk
Carry the day's cash, not the trip's. Keep the rest, along with your passport, wherever you are staying, and keep a backup card in a separate place from your main one so that a lost wallet is an inconvenience rather than a crisis.
Small notes matter for a practical reason as well as a discreet one. Producing a large-denomination note for a short taxi ride creates a change problem and displays more cash than needed at the same time.
Use ATMs in well-lit, busy locations, and malls are ideal for this because they combine banks, machines and licensed money changers under one roof with security present.
Ordinary street sense
Keep your phone in your hand rather than in a back pocket, and be aware of it in crowds and on packed trains. Use a bag that closes properly. Do not leave belongings on the back of a chair in a busy restaurant. None of this is specific to Manila, and all of it applies here.
Traffic itself deserves respect. Crossing roads in a city built around heavy vehicle flow requires more patience than in a pedestrian-first city, and using footbridges and marked crossings is the sensible default rather than an inconvenience.
Weather as a safety topic
During the rainy season, weather is a more likely disruption than crime. The national weather agency operates a five-level tropical cyclone wind signal system, and signal levels are raised for specific areas rather than nationally. Higher signals bring automatic suspensions of classes and government work, and private businesses and transport generally follow.
The practical habit is simple: during the rainy months, check whether a signal is up for your area before committing to a day's plans, and treat heavy rain as a reason to be less ambitious about crossing the city, because road flooding affects journeys before it affects anything else.
Documents and backups
Keep a photograph or scan of your passport data page, your entry stamp and your travel insurance details somewhere you can reach without the original. Email them to yourself so they survive a lost phone.
Carry a copy rather than the passport itself for day-to-day purposes where possible, keeping the original secure at your accommodation, and take the original when you have a specific need for it.
What this adds up to
Nothing on this list is dramatic, which is the point. Use official transport, keep money and documents managed sensibly, ask locally about your immediate area, and pay attention to the weather in the rainy months. Applied consistently, that covers the overwhelming majority of what a first-time visitor to Manila needs.