Welcome back:
If the battery is good, I need you to confirm there are no loose or corroded terminals on or from the battery. When you say the lights dim or go out, that is either a weak battery as Ken mentioned, a loose or corroded connection, the starter is bad, or the engine is locked and the starter can't turn it.
What I need you to do is check if there is power to the starter. On the starter you will have a large wire straight from the battery positive. Is should have 12v all the time. There will also be a smaller wire which only gets power when the key is in the start position. I need you to confirm that is what is happening. If it is getting power, remove the starter and have it bench tested. Most parts stores will do it for free with hopes of selling a new starter. If it passes the bench test, see if you can turn the engine at the crankshaft using a socket and breaker bar to confirm it can turn.
Here is the link I was referring to. Take a look through it and let me know if it helps or if you have other questions.
https://www.2carpros.com/articles/starter-not-working-repair
I need to know one more thing. You replaced the battery, so it should be good. When you said the lights go out, does everything electrical stop for a short time and then start working again? If so, take a look through this link:
https://www.2carpros.com/articles/everything-goes-dead-when-engine-is-cranked
Let us know what you find.
Take care,
Joe
Saturday, November 2nd, 2019 AT 11:29 PM