Road bikes are faster in general for several reasons:
1. Slick tires offer less resistance
2. They are lighter, thus going faster is easier
But, the ease of climbing is also dependent on the gear ratio. Very expensive road bikes will not have very low gear ratio and as such will be harder to pedal. This should not be a problem for you as you are on a budget.
Most brands (even good ones) will have entry level bikes. My suggestion is that you go to a bike store and explain your situation, do not buy another bike from Target or Wallmart etc. Sales people do not know what they are selling, to them a bike is a toy not a means of transportation or a serious sport item that need to be the right size etc.
Specialized and Trek are both very reliable and good brands, most bike shops will carry those in stock, ask for entry level bikes and go from there, take their advice as they are usually good experts.