You can book your flight directly into Cusco. But it may be cheaper or you may find better flight times by booking a flight into Lima and then getting another flight with a national airline to Cusco. In all cases you have to go through customs and recheck your baggage to your flight to Cusco
Good carriers are:
Latam: https://www.latam.com/en_us/
SKY: https://www.skyairline.com/english
Starperu: https://www.starperu.com/es/
Viva Air: https://www.vivaair.com/co/en

I would recommend anyone going to trek or climb in the Peruvian Andes to not stop in Lima first. Visit Lima after you have done your trek or climb, the reason for this is that you will be much more susceptible to getting sick in a large, contaminated city as opposed to the dry more sterile environment of the mountains.
There are some cool things to see in Lima that may merit a couple days, but if you don’t have much time for the trip, then your time is better spent in Cusco as there is much more to see and walk around being more tourist friendly and interesting. In Lima the Plaza de Armas with the Catherdral / Museum on the East side where Francisco Pizzaro is buried along with many of the first viceroys of Lima is nice. The Convent of San Francisco with its catacombs and Parque de las Murallas close by. The Barranco district is charming for its colonial architecture and atmosphere, as well as the restaurants.

Huaraz is reached by an 8 hour bus ride from Lima. While there are good bus companies like Cruz del Sur or Movil Tours who have their own bus depot in Lima, it is better/easier to take a bus from “Terminal Terrestre Plaza Norte” which is only a 15 – 20 minute taxi ride from the airport. Because this is closer to the northern exit of Lima and will reduce the time you are sitting in traffic.

Getting to Arequipa is quite the same as your options for traveling to Cusco. Either a very long bus ride or a flight of 1 hour with the price being a bit more. To maximize your time and energy it is obviously better to opt for the short flight.

Taxis from Cusco airport to the city center for a tourist should be 10-20 soles depending on where you are going. For any ride around the city center it should be about 5 soles in Cusco or Huaraz.
Our trips include private transportation, from shuttles to 4×4 Trucks, depending on the group size.

Bus travel in Peru is quite cheap, easy and comfortable compared to other countries. All of the good companies have seats that recline 160° almost flat or even 180° flat. The cost of carrying a large duffel full of gear is minimal. But in my opinion the longest bus ride that is worth it is up to 10 hours, beyond that you should save yourself the time and just fly.