This is probably one of the most interesting and important subjects that universities failed to include in your curriculum. First I will define a Career Mentor:
A Mentor is a professional in your target field that has accumulated decent experience and is willing to help you bridge the gap between theories you learned in college and the real world applications. He will also do that for free, because he just likes to help you for who you are.
Yes you are basically trying to land an opportunity of being taught real things that can add value to your life without even paying for it.
Most of the successful people had a Mentor and you will hear most of them talk about this man that helped them so much when they were young (think Ben Graham - Warren Buffet). It is very important to have one. I have recently landed one so I know something about how it goes.
Please note that your Mentor could be your Manager at work, this means you have already landed your dream job + you are getting supported by your superiors, Perfect! If not, then this is for you.
1) Basics:
First of all you have to know what is your target profession and the niche you want to specialize in. Once you have figured that out, it is time to go out in the real world or linkedin, the real world is preferred in this case.
2) Linkedin:
You will use the search tools to find experienced professionals, 5+ years of experience and start cold contacting (5 years because they are still young and can understand your position. where they were not so long ago).
You will be writing brief messages saying: "Hello, I am a recent Graduate from XYZ and I am very interested in learning more about Refining Crack Spreads. I have a passion for Energy Markets and have a couple of questions on how to evaluate the Throuput Production of Philips 66. Can you please help me figure it out?"
This will turn into a longer conversation and this is where you can develop a long lasting relationship, eventually asking them to meet up for a coffee and discussing topics of your interest and who knows, maybe they can help you in your job search.
3) Networking events and seminars/conferences in your region:
Handshaking someone and introducing yourself politely has much more influence on someone's opinion of you than a message on Linkedin from as a stranger ("who's this guy").
Seminars typically focus on a specific topic and therefore are targeting the niche segment you want to specialize in automatically. It will be full of professionals in your target industry that might not have Linkedin or might have simply ignored your cold call (don't worry they won't remember it).
Please always remember to wear a Business suit with a simple tie and shirt, be clean, no fancy watches and nothing flashy, you want to be perceived as Neutral from an appearance standpoint so you can impress with your enthusiasm.
"There is no second chance to make a good first impression", make sure you know already how to introduce yourself properly and your elevator pitch.
As soon as you have introduced yourself, start launching a wave of very smart and impressive open ended questions that will keep talking and talking, make them feel understood with body language. This will lead them to feel good being around you. You can also use what has been discussed during the Seminar as a starting point to get them to start talking, ask them about their opinion of what has been discussed.
At this point, they might give you their business card and tell you: "let's stay in contact" or if you are lucky: "Hey I think we're looking for an Analyst why don't you apply". If they don't offer you a job it's completely ok, the focus here is to have a contact you can send all your questions to and get answers. If they forget to give you their business card and you feel that they appreciate you, feel free to ask them for it.
Please also note that your mentor, might give you a couple of assignments to do for him. Remember this is a 2 way relationship. These are great opportunities for you to show him what you got.
Even if he does not hire you in the future, he might recommend you to hiring managers when they ask him whether he knows anyone decent for XYZ role.
& Even if he does not recommend you, remember the goal was to have someone who can advise you and help you with your questions.
Please remember to never tell them: "Can you be my Mentor", this kind of relationship develops very naturally and they know they are mentoring you when they are answering your questions, no need to highlight it.
Try to go about it more like a friendship than a student-teacher relationship but always make them feel that you appreciate their help.
Hope to have helped here, feel free to shoot any questions or to add your opinion on any of these points,