At level 1, what capital cities can I go to?
If I create a new character and want to go to the capital cities, which ones can I successfully make it to in the Warlords of Draenor (v6.1) expansion?
Additionally, how can I make it to these locations?
For instance, can I take a mage's portal or be summoned to Ashran at level 1? If not, what are the requirements to be able to go there (or other capital cities, for both factions)?
To start, as a level 1 you should be able to make it to and from all of the major racial Azeroth cities with little issue. Generally you will find a boat or other means of "long" transport (e.g. the tram from Stormwind to Ironforge).
Mage Portals & Warlock Summons
Additionally there are no level requirements attached to the Mage's portals or the Warlock's summon ability. This means, assuming you have a Warlock that is at least level 42 (when the spell is available) and 2 willing friends near him to help him summon you, you can literally go anywhere that the Warlock is willing to go first. If you have a Mage friend, you can go anywhere he can make a portal for. You can find the portal locations available to a mage on the Mage Portals wiki. As you might assume these portals include your standard capital cities of that Mage's faction as well as places like Shattrath in Outlands, Dalaran in Northrend, Vale of Eternal Blossoms in Pandaria and to answer your specific question, his faction's city in Ashran (Stormshield and Warspear respectively).
Finding a Mage willing to make a portal for you is your easiest means of travel as it only requires one person opposed to a Warlock basically needing himself and 2 additional members to both summon his ritual stone and then to help summon with said stone. With enough money you should be able to talk a Mage into doing this for you in trade chat, which is opened anytime you are in a major city.
Capital to Capital Travel
They've made the capital cities for each faction incredibly connected, whether you take a boat from one to another or often they simply put portals. They've also set up portals to the expansion zones directly from the cities, however these portals usually don't appear until you've either completed a certain quest (of the appropriate level) or they could potentially be level locked.
As for the idea of getting to an opposing faction's capital city, this is potentially more difficult and will require some foot-travel in most cases as you would have to make your way to the closest place that your faction controls near your destination then go by foot or mount from there to your desired location. I'm not sure what you'd hope to accomplish if this was your mission as a level 1 however, it would likely be chain deaths and would take hours. Naturally any portals to the opposing faction's city will either simply not appear to you or will be faction locked.
I didn't go into any real specifics, I'm sure you can find the right boat for the job if you're plan is to start a character in one place and then travel to another to level. If I knew more about what you were trying to accomplish I might be able to get you more specified information.