You have to acclimate the fish to the water temp/chemistry when doing water changes- especially large water changes. A sudden change in temperature will put them into shock, which tends to be fatal. I am sorry you lost your betta..
Fins will grow back, but if he isn't swimming properly then I would recommend replacing him. As new fish owners it's better to start off with a healthy fish, as treating a sick fish can be stressful.
When doing water changes, place them in the cup you get them in.. and when returning him to the tank, float the cup in the water for about 15-20 minutes. Every 5-8 minutes or so remove some water from the cup (dump it in the sink or in a cup, don't place in the tank) and replace the water with water from the tank. I would say no more then 25% of the cup water should be changed at a time when doing so.
This way he will adjust to both the temp and the chemistry change.
Keep up on the weekly water changes to keep him healthy and happy- depending upon size of tank and whether it is filtered, the schedule will be different.. but weekly is still a must.