If Sirius A was to take the place of our Sun, how would it effect our climate?

Oceans would boil, the surface of the earth would be superheated, then all the gas would slowly be blown off into space as our magnetosphere wouldn't be able to handle the radiation output. So first we become something like Venus, then something like Mars, then something like the Moon.

Sirius A is about twice the mass of our sun, with about 6 times the output... 

If you think global warming is bad now... try it with a star 600% more output than we have now.  I'd give land life forms a few days, and the life in oceans a couple of months before it was all gone.

Since Sirius is a star, it only shines at night.    The Earth days would be dark, but the nights would be light. 

The oceans would boil and we'd end up like the surface of Venus.

It is nearly double the size of the sun, so a direct replacement in the sun's exact spot would increase the gravitational pull, but probably not by a lot. It's 25 times brighter so we'd have extremely bright days, probably wouldn't be able to go outside at least without heavy protection for our eyes and skin. Being a white hot star compared to our yellow sun, it's an average of 2,000K hotter, which will make it hotter here.

It's bigger but that's not an issue.

It is 25 times more luminous than the Sun, meaning it would get very, very hot, boiling off the oceans and water, and almost everything would die off! Except for maybe roaches and tardigrades.