Q:

The right way to escalate things

I think it's a somewhat common situation in the workplace but one I simply can't master.
My boss tells me to turn to colleague A for some information. So I turn to colleague A and propose a meeting. Colleague A ignores me repeatedly although I contact them several times, using different methods (arranged meeting, email, spontaneous call) and trying to make it as easy as possible for them.
I try to get the information from other people, but there's only one source of this info, colleague A.
I don't think there's a solution apart from escalation to my boss. But how do you escalate without souring your relationship with A and with your boss who values self-reliance and subordinates who solve problems on their own?

A:

without souring your relationship with A

Which relationship? You tried your best and were ignored... so there is already a big issue with your relationship.
Most important in such situations: don‘t be the weepy colleague running for papa (your boss) to help you out.
Do one more trial to get the information but be sure to have a paper trail:
Write something in the sense of The following in a mail:

Hi A,
Boss C told me to do X and to get the needed information for this action from you. 
Can you please give me the informationen via mail or make a suggestion for a meeting in the next days to tell me what I need to know?
Thank you in advance.
  Me, myself and I.

Make sure you carbon copy your boss. 
Ccing somebody who gave you a task is a common thing to do, but it does two things:

It brings the ball into A‘s field and 
makes him aware that the boss knows this.

If he still ignores you there are two ways to go -depending on the urgency.
The first thing would be to send a reminder mail a few days later:

Hi A,
I am sorry to ask again, but I still did not get any response from you regarding my mail from 3 days ago.
  Can you please consider to give me the information asap as I am stuck with my todo.
Best regards
  Me again

Of course again carbon copy your boss.
If A still does not react then You need to go to your boss. But now you have your paper trail. Tell him that you tried to call and make appointments as well and that nothing helped.
Then it is his responsibility to do whatever is needed.
The second way to go would be to omit the second mail and go directly to your boss. As said: this depends on urgency and how important this thing is for your boss.

