Home is free and you download it directly from the Icon on your XMB menu.It's basically Sony's answer to Xbox Live's Avatar characters. Except you can interact and move around unlike the 360's Avatars that just do what they're program to do. You can go shopping, visit cool homefront pages, play mini-games dedicated to many games like WipeOut, Killzone 2, Resistance, Resident Evil 5, LittleBigPlanet, ModNation Racers, etc etc. Tons of promos to win free gear for your avatar and a movie theater where you can watch trailers, interviews with programmers, and even PS Nation Podcast in the near future.
The only time you ever have to pay for something on the PS Home is if you want a special costume, clothing item, furniture, or a new theme for your apartment. Think of it like the SIMS but you are rewarded with many cool things just by browsing around and even certain games. i.e. Batman Arkham Asylum unlocks the Bat Cave for you to hang out in!
Trophies are Achievements essentially. I'll have to see if there is a way to turn off those alerts.