Q: When i drive the car's dash indicator for the Hot-Cold after 10 mins of driving goes to the hot setting ? Where should I start?

Car broke down on me a couple of weeks ago. It was overheating to the point where smoke was coming from under the hood. Eventually as I was driving and the car dash lights start going all crazy then the car dies as I'm driving even the power steering and breaks were hard to press. after that we had it towed to my house. At first we assumed it was the alternator, I popped the hood the next day and i had noticed that the battery connectors and posts had a lot of acid on them i figured this was why the car had died so i purchased a battery cleaning kit with new connectors charged the battery for about 30 mins or so then continued to check the antifreeze and tank noticed that the cap for the tank was broke (not allowing pressure build up) so i replace that part. check if there is fluid in the radiator - none. fill with antifreeze massage the tubes coming from and going to the engine from the radiator, car starts but still over heats?
My car has an automatic transmission.