All new tyres need a running-in period. To run in your tyres, you should drive at moderate speeds during the first 125-200 miles, in order to improve long term tyre performance. Until you have adapted your driving style to the new tyres, rapid accelerations and sudden braking during the first miles are not recommended. If your old set of tyres was considerably worn, be aware that the your vehicle will operate differently with the new set, even if the brand and the characteristics are identical.
When you check tyre pressure, you should also check treadwear. TWI indicators are branded into the sidewall and allow the driver to locate wear check-points. A tyre should never be worn down to the bars beneath the treads' surface and wear should be evenly distributed.
Tyre wear leads to a loss of grip. The more worn your tyres are (indicated by shallower grooves), the longer braking distances will be, especially on wet roads. The risk of aquaplaning will also be greater.
Tyre position (geometry, alignment) and your driving style can cause your tyres to wear differently. To evenly distribute wear, rotating front and rear tyres every 3000-6000 miles is recommended.