I've stayed in Edinburgh 4 times with each a wonderful experience.
Two that stand out for great locations:
We stayed at The Hub at Premier Inn off of Princes Street. To us, it was centrally located for the tram, shopping, and places to explore in New Town. The price was reasonable, it was super clean, and they will also store your luggage for you until you are ready to check in. From the location, it took only a few minutes walking time to reach the Royal Mile.
The other, was a self-catering apartment beside the castle. It was all about location. I loved having my coffee in the morning, looking out at the castle in one direction and Arthur's Seat in the other. It was convenient to open the door and step out into the action without having to take a taxi/bus/train to get to the center of town. You can look on many accommodation websites, i.e. booking.com or Airbnb.com.
As for Glasgow, we stayed in a boutique and trendy hotel (Z Hotel), which was near the pedestrian area and very close to the bus and train station. They had a great wine and cheese in the evenings and a tasty (free) breakfast in the morning.
As for weather, I tend to travel in the shoulder season. My favorite months to travel are Sept/Oct. Prepare yourself, the weather can change in an instant. Wear layers, take a rain jacket and umbrella, and invest in some good water resistant or waterproof shoes with non skid soles. I made the mistake of not bring the right shoes one time and slid all over the wet cobblestone streets of Edinburgh!
Good luck on all your travels!