What is the best way to level up Frames in Warframe?

Since Draco has been brutally murdered by DE the answer to this question has changed. People have found various nodes as replacements for Draco. The most popular are currently;

• Berehynia, Sedna
• Hydron, Sedna
• Akkad, Eris

I personally prefer Berehynia as it has the highest enemy spawn on the star chart, a distinction that was previously held by Draco. This means that time-wise with the appropriate squad, Berehynia is currently the most effecient way to level stuff in warframe at the moment.

Try Orokin Derelict Defense (ODD for short). Gives lot's of experience if you do a great amount of waves, it was easy to upgrade my nekros this way. Also, try Interception, this mode spawns lots of eximuses who give about 600-700 xp each.

spy missions, mainly tier 3 spy missions. since its a spy mission, you dont need to equip all of your weapons, which will help you level your warframe up faster and each data hacked will give you a great amount of exp.

For frames, I'd recommend a Tower 1 Defense. They don't take much work and there are always a lot of people willing to do them. Orokin Derelict Defense is ok, but it's much more difficult now due to the TAR Moas.

The biggest thing to keep in mind is that more enemies = more experience. Interception is great for getting tons of enemies but it can take a very specific group of people (draco currently "requires" 2 excalibers, a trinity, and a rhino) so you would be hard pressed finding a group that lets you level a frame. The interception method is great for leveling weapons though, so don't be afraid to throw on something low leveled to snag some extra experience.