Who doesn't remember their first solo kill. IMHO the first solo kill in Eve is the most intense gaming experience ever.
It starts with the hunt for a ship that looks possible to kill. Once found then the heart starts pumping as the adrenaline kicks in. You power towards the enemy trying to activate modules that seem to take forever to turn on. Finally you have point and your weapons are blazing with all their might. Tunnel visions sets in as you notice that his shields and armour are dropping faster than your own... and finally you see the explosion.
For me my first solo kill was an older character called Johny Trigger. He was in a Rifter and I was in my shiny new Stabber that I could barly use the 220mm Autocannons on. At the time I had been recently kicked out of the Minmattar Militia (friendly fire) and I was now flying in a wannabe Mercenary corp called Frost Company Mercs. The corp was based out of the Uisper system in Metropolis low-sec space. Mostly we just ratted and docked up whenever anyone was in system with us.
Anyways I had just gotten enough skills and money to put together a Stabber so I headed of to Hek to go and put one together. When I came back into my home system there was one other pilot in local so I thought why not, and I tried to hunt him down. After a few minutes ineptly using my directional scanner I found him at a belt so I warped there at zero. His Rifter was about 80km off of me when I landed so I started powering towards him. I mostly expected him to bail but instead he powered towards me and the battle was on. I aggressed first, taking a hit on my sec status. However he made the mistake of trying to orbit me at 5km, within the tracking and range of my 220mm 'Scout' Auto-cannons. He probably wasn't expecting me to be armour tanked and fitted with scram-web (WTF kind-of fit is that!) and he soon went down.
All I really remember of the fight was feeling like I was having a heart-attack my heart was pounding so fast.
Icing on the cake was killing him again several days later, this time in a Thrasher :)