As Eric said above, the majors want to see, Part 121 turbine PIC time. Most also want a college degree.

There are two paths to the majors, the military route and the civilian route.
With the military route most go right to a major because they are a "know quantity", in other words the majors know that they had top level training and they have met known standards.
In the civilian route you come to the interview with a resume of a lot of hours and what and where you have flown. To them your training is unknown. So put yourself in the position of a interviewer. You are hiring someone to work for the next 30-35 years for your company. You need to know that all the money and time the company is going to spend on training the individual will pay off. As a friend who is an interviewer for United told me, " we are hiring future Captains, we need to know you can be a Captain."
United went through a period in the 80s when they hired a bunch of lower time pilots. When it came time for these pilots to upgrade to Captain, many had problems passing the checkride because they lacked the experience of being a PIC in a 121 operation. That is why the majors want to see Part-121 PIC time. They know you have passed a FAA Type rating check-ride,
so now you are more of a known quantity.

Another thing to consider: The airline industry goes through cycles of hiring and no hiring. All the airlines have published minimums for hiring, but they really don't mean anything.
What really matters is what we call "competitive minimums". All airlines consider real world experience above all other experience (remember that 121 PIC thing). So competitive minimums means, what is the level of experience of the people who are currently getting hired. For example, when I got hired at United in 2001, the pilot with the lowest time in my new hire class had 3500 hrs, with over 2000 hrs of Part 121 turbine. The competitive mins change with the current conditions of the airline industry. Right now there has not been much hiring at the majors for several years. So when hiring starts up again, the competitive mins will be very high because the is a large pool of pilots out there with a lot of experience.
So imagine you are the hiring guy at United, you get thousands of resumes (back in 2001 United had 10,000 apps in their system) which ones are you going to pick to interview?
The furloughed airline guy with 10 yrs part 121 experience, the regional Captain with 4000hrs of turbine PIC time, military guy with 2000 yrs , or the guy with 2000 yrs working as a CFI training new pilots?

I'm not trying to discourage you, but if a job at the majors is your goal, you need to know what it really takes to get hired.

1- If you have a certain major as your goal, learn what they require to get an interview. For example, at Southwest you will not get an interview without a letter of recommendation from a
Southwest pilot. (at all majors a letter of rec will makes you more competitive)
2- Start building Part-121 turbine time as soon as you can.
3- get a degree

All that being said, having airline pilot contacts and friends can get you a little ahead of the competition. EAA is a great way to do that.
Just having a lot of hrs will not get you in the door at the majors.
Good luck!