When I added the amount contributed to the HSA, the amount of taxes I owe increased? Should it have stayed the same or decreased?

The amount you owed should probably have increased, and then later decreased.  

Entering information into your return regarding HSA contributions is a two-step process.  Each step is likely to cause the refund/balance due number to change because TurboTax constantly updates the calculations as you go through the return. 

The first step occurs when you enter your W-2 that includes contributions to the HSA.  After entering the information from your W-2, the amounts contributed to the HSA are included in your income (thus causing a higher balance due or lower refund). 

Then, the second step is to verify your eligibility to contribute to the HSA due to coverage under a High Deductible Health Plan.  This results in the HSA contribution being removed from income since you were eligible to make the contributions.  Therefore the balance due amount will drop or the refund amount will increase. 

To go directly to the section of your return to verify your eligibility, use these steps:

• On the top row of the TurboTax online screen, click on Search (or for CD/downloaded TurboTax locate the search box in the upper right corner)
• This opens a box where you can type in “hsa” (be sure to enter exactly as shown here) and click the magnifying glass (or for CD/downloaded TurboTax, click Find)
• The search results will give you an option to “Jump to hsa”
• Click on the blue “Jump to hsa” link and enter your information