The easiest and least complex solution would probably be something like this.
This would be the first "station" in your minecart transit system. To use it, you would place a minecart on the unpowered piece of track and get in. From there, you can press the button, and because the button is technically occupying the block right next to the rail, it will power it, launching you to the right.
The reason this works is because there is a block directly next to the minecart when it is placed down, and powering the powered rail will start the minecart. You will notice that if you were to remove the block and press the button, nothing would happen. This is because the minecart "doesn't know which direction to go". This is important to note when building stations based on powered rails.
At a stop, you could simply do something like this. It is identical to your first station, except for the button being placed differently. Your minecart will stop on the unpowered sloped rail because powered rails act like a brake when unpowered. However, if the button is placed where it was on the initial station, it is hard to press due to being on a slope, so you have to raise it up and use wire to connect it to the rail. If it's your station, you can get out of the minecart; if you want to continue to the next stop, just press the button.
This can be modified into a two-way station though the use of detector rails. This is what your middle stops could like like in a two-way transit system:
If you are approaching the station from the left, your cart activated the detector rail, turning the unpowered track on, and stopping you on the right side of the block (and of course, vice versa coming the other direction). However, if you approach the station too quickly using this technique, your minecart won't stop properly and it will touch the second detector rail, powering the rail you are supposed to stop on, and you'll keep on going. Just don't go overboard with the powered rails.
Williham Totland came up with an even simpler two-way station by using a dip instead of a rise, which I now prefer.