So I bought my 2005 Mazda6i 4 cylinder. I am not car savy at all, well. Almost a week ago, I was driving down the road and my alternator had gone out. The car shifted HARD into 3rd gear, the Traction control came on, ABS came on, I never got a check engine light or battery light come on. So, I drove it home, and had the alternator tested and it was bad. Now I've never had any problems with my transmission. EVER. So, I put the new alternator in, my headlights were working and everything so. I go to drive to work and... it shifts immediately from 1st gear into 3rd gear REALLY HARD. I thought that maybe it needed some transmission fluid, (yes it is an automatic) so I added half a quart. Now, when I stop at a light it idles, VERY ROUGHLY. So I can get it to go from 1-2-3-4th no problem and manage to hold it and sometimes its okay. But most of the time it'll drop from 4th into 3rd REALLY hard, or bounce from 1st into 3rd really hard, and the radio will go out and the headlights I noticed now... don't go out.. but they slightly dim on and off. I have no idea what to do, I called mazda customer service they told me to go to a dealership and have it tested, but thats an arm and a leg and im a college student lol = no money. So, I went to auto zone today and got my battery and alternator tested and they said there was no problems there at all everything tested fine. Now I don't have any check engine light or anything, but i'm trying to figure out what to do, or what could possibly be the problem((

 

You could have more than one problem or it could be one problem causing all the symptoms.
First make sure that the battery cables are not corroded and properly attached.
Second make sure you have good ground connection to the body and to the engine and transmission.
here is a good video to get started: