Bus from Mexico City

That is a great idea! We did something similar in 2005. We stayed in a very reasonable hotel 2 blocks from the zocalo on Avenida del Brasil, 3 minutes from the cathedral. We took a TURBUS tour for one day in order to get a bit more familiar with the city and after that we used the METRO which is extremely efficient, inexpensive (buy several boletos at the same time)convenient, easy to navigate and we felt very safe using it during daylight hours.Do not miss the Museo de Antropologia as it is considered one of the best in the world and I do agree after spending 2 days exploring it. If you have a chance to see the Ballet Folkorico at the Palacio de Bellas Artes besides the Alamada park. Castillo Chapultepec , museums, Xochimilco , San Angel, Coyoacan ,all those sites easily reached by Metro and local trains.As you can see Mexico City is a wonderful place to explore.
We took a bus to the village of San Juan de Teotihuacan and it is worth to spend the night there in order to arrive early at the site Teotihuacan and avoiding the huge surge of people arriving by the daily tours offered from Mexico City. After visiting years ago the pyramids at Giza ,Teotihuacan was still able to take my breath away........... it is magical!
We went to Guanajuato from there but visited Morelia , Patzcuaro and Uruapan on a different trip where we travelled on either Primera Plus and Parkuni buses that were very comfortable .
There was a discussion on this board of a nice play to stay in Morelia and a deluxe bus service Zihuatanejo-Morelia.
Have a great trip and dont forget to get yourself a good guidebook such as Lonely Planet as it is full of good info.