What else can I do or replace or look at in my vehicle for a reason why it still won't start? Had ignition and starter replaced but it still won't even turn over.

We recently had this car shipped here to Wisconsin from Arizona and when we
got it, it would only start every 10-20 key turns and before it would start, it
wouldn't do anything, no noise, no crank, just dash lights came on and then all
of a sudden it would start. Was told by mechanic shop that it wasn't the starter
but that it was the ignition with security issues and that the dealership was the
only place that could do it. I took it to a Buick dealership and they replaced the
ignition and gave me two new keys. Then they called and told me it still
wouldn't start and that it needed a new starter. I told them hell no on the
starter and went and picked up the car, brought it home, and replaced the
starter myself but it STILL wouldn't start. I have no idea where to go from here.
The battery is practically brand new (4/15 sticker on battery) , it died when we
towed it to my house (long story) but we jumped the car after replacing the
battery and this is when nothing happened. The weird thing is tho, we applied
the jumper cables, put the keys in, turned the ignition and nothing happened.
We reapplied the jumper cables a few different times and still nothing
happened. I removed the keys and all the dash lights stayed on so we
disconnected the battery and the dash lights turned off. We reconnected the
battery and all the dash lights came back on again without the keys being in
the ignition. Very strange. I don't really have a clue where to go from here but I
do know I don't want to take it back to the dealership. I mean I was told the
starter was fine, took it the the dealership, spent $550 on the ignition, then
they wanted to replace the starter for $350. What if I had paid them to do that
and it still wouldn't start? What else would they have replaced and for how
much more money? Does anyone have any clue where to go from here?