Would you be more likely to buy new gear out of state if there was no sales tax

I use Turbo Tax for state tax also. Must be different state to state.

I could not find anything saying it is our responsibility to total up our purchases and somehow pay at tax time.

Bill, Google "sales and use tax" (with the quotes) plus the name of your state. That will lead you to an informational pdf on the subject issued by your state tax authority.

We buy stuff on Amazon etc...with no tax, but must keep track of all these purchases and somehow pay the tax to state we reside in.

BTW, as you are probably aware most of the major online retailers make it very easy to print out a summary of purchases one has made from them during the course of a calendar year. Under My Account/Order History or some such link.

Best regards,
-- Al

I have done this tactic in the past. Absolutely!

After reading per Al's directions all I can say is....what?!  The secret is now in the open. First I have ever seen or heard of this fact. Seems very, very few know of it or pay taxes based on it. 

One more reason to move from my tax heavy state which will happen soon. 

I suppose next we will need to pay tax on used goods sold privately such as audio gear. Please don't tell me......I shut my eyes and ears on this one :)

AFAIK charging tax on goods bought in another state is against federal law.  States charge instead the same tax naming it differently.  For instance my state, Illinois, charges "Use Tax" which is tax for using goods bought out of state.  Since I don't want to investigate what was bought with or without tax - I elected option to pay flat fee proportional to income.  This option can be selected only if there was no big items bought without a tax.  Out of State tax can be very messy.  Some stores charge tax for items sold to another state illegally, because they have to be authorized, by the other state to collect this tax.