I installed a new battery on my CT3.....   

Turned the key switch to on (both positions), but no lights came on and I heard distinct "buzzing" sound.   Any input as too what I should look at first?

Responses have been posted in your "CT3 purchase" topic where you originally asked for opinions... it's okay to start a new topic with a specific question.  Just know that it can get confusing when someone starts asking the same question in multiple topics around the same time.

Sounds to me like a short circuit of the blinker relay if it has the bigger 3 wire relay of which one wire was earth. I'll go look at the diagram now i've opened my mouth haha, but have heard that noise before years ago. Unplug it for a start then let us know.

Yep here it is. Not 3 wire as i thought but this is the type anyway. So if it does end up being 'that' there must be a short somewhere along the brown white to the switch--or i'm barking up the wrong tree again!!. 

I might be conflating some theories here (sorry!), but if it is the flasher relay buzzing, then maybe wear insulating gloves before handling it - someone I know of got a nasty shock and burns when he handled a flasher relay in similar circumstances. Any energy that it is storing needs to be discharged.

Experts, please now step in to confirm what points I have made are valid and what are rubbish! 

Sounds like a big condenser GR8UNCLEAL.

It is the blinker relay...... thank you for all of the input.

Yay!!. or not a short after the relay just loading it up with nowhere to go?. Good news anyway.

Picked up a relay from Pobits on ebay......   fit right where the OEM was installed and works good.