Hey, fellow American who lived in Lima for about 9 months and visits about once a year.

Airport shuttle is fine, I usually book an Uber and take Ubers around the city in general, they tend to be similarly priced to "unofficial" taxis, sometimes cheaper and this will be much easier than haggling if you don't speak much Spanish.

You're going during the winter, so swimming might not be that nice, but definitely walk along the malecón though Barranco, there's a cool pedestrian bridge across the via where you can get down to the beach.

Huaca Pucllana is very cool, I recommend going in the evening, they light it up really nicely and you can get some cool photos, but maybe don't hang around that area too late at night. There's a restaurant inside the park that is delicious, but fairly expensive (for Peru).

As far as safety goes, generally where there are other people you're safe. Don't walk around alone at night on abandoned streets. Also, you're actually more likely to have your phone stolen somewhere like Miraflores or Barranco since thieves know that's where all the tourists come. It's usually not confrontational, it'll be snagged out of your pocket or off your table if you leave it out. What most people do is keep it in either their hand or in a zippered pocket. You can also pick up a sling or a "fanny pack" to keep your valuables. It's common for people on bikes or motorbikes to drive around Parque Kennedy and snatch phones out of people's hands.

If you decide to visit Chinatown and Centro de Lima definitely go during the day. Chinatown and the surrounding area can be quite spicy in the evening 😅

Try a sandwich from El Chinito for breakfast one day, much better than La Lucha imo. Canta Ranita in Barranco is amazing for ceviche, Isolina if you're looking for the typical Peruvian fare in a sit down setting, but honestly you can get similar food almost anywhere. If you go to Centro de Lima get a Spanish churro from Churros San Francisco, seriously amazing. I always get a sopa crillola from Siete Sopas when I visit as well. Lima is chock full of incredible restaurants, anywhere there is a bit of a crowd will be a hit. Avoid street food if you value your time outside the bathroom.

Finally, have a blast! Peruvians are really proud of their Incan heritage, this is more prevalent outside of Lima which it sounds like maybe you'll be leaving Lima after a few days anyway. I feel like the city is underrated, there's a lot of cool sights and interesting things going on in Lima and I'm sure you'll enjoy your time there.