I agree with everyone else who is suggesting that life for the INTP (unless the INTP has been fortunate with a supportive family and environment) begins at college.
That's the place where your intellect will finally really be engaged, and it's where you are most likely to FINALLY start meeting a decent number of people with whom you can identify and who identify with you. (At that time in my life, I had no idea what an "INTP" was... but I did finally start making friends who understood me.)
You will also probably start studying things and doing things that will help you decide what direction you want to go in the next years of your life. Usually childhood seems full of wandering around, being bored, and wondering what happens next.
My childhood was pretty lonely, all things considered. I lived in a rural area with few friends nearby to begin with, really, and my family situation meant I wanted to be alone much of the time. I either hid in my room drawing, reading, listening to music, or doing something craft-y, or I'd be outside exploring the fields, climbing trees, digging through the garage, making things, or riding my bike for miles... usually alone. I really liked to wander; but I remember being very lonely and wishing I had someone to understand me. No one in my family understood me -- this ended up being a very big issue. (It also created issues later in life when it came to building new relationships.)
I remember getting picked on in school (through part of high school, then people got bored and left me alone). If I aced a test, I was made fun of; if I didn't get the highest grade, I was made fun of; and people would steal my stuff and hide it. All the typical shenanigans. I did finally make friends in the music program in high school, but still no one really understood me even if I felt accepted.
Besides not being understood, I had few (well, almost none) who shared my interests and were intellectually on my level. This also led to me being alone.
I also was extremely shy, socially, and I didn't have much confidence or skill in expressing myself or my emotions. I was really good at music and complex topics (or traditional "geek things"), but most people didn't want to discuss that sort of thing all the time, so once conversation became more conventional, I no longer knew what to say.
Finally, I grew up in a religious environment that really tried to pit my intellect against my virtue -- i was told I was a good person if I believed and did certain things, yet I didn't agree with all those beliefs and/or behaviors. Also, because INTP is so big-picture and quick to notice discrepancies/illogic, I was always very good at seeing how I failed to conform to my "faith" and then mercilessly depricating my self-worth because of my taint. I couldn't act rationally and consistently all the time, and I thought I was "bad" because of that... and I had no adults who thought on my level who I could talk to about it.
All in all, my teen years were the worst of my life... aside from the times I spent alone, I think. I have some great "alone" memories, and memories of books I read and topics studied, and places I explored... but honestly college was a vast improvement.