Travel with luggage.

I have a few questions:

-Is the NEX a good way to get to Tokyo, with luggage?(One suitcase, and a small bag)
Also, what does "reserved only" mean? Will I be able to reserve a seat when I just get there? If not, what is the next best option?

-From Tokyo to Kyoto, is a green car on the Shinkansen spacious enoguh for my luggage?

-From Kyoto to Osaka, how much would a Taxi cost?

-What is the best option to get to Kansai Airport, from Osaka?(with my luggage)

Thank you!

Is the NEX a good way to get to Tokyo, with luggage?

Depends where in Tokyo you are heading to. But the NEX has ample of space for large luggage.

Also, what does "reserved only" mean?

It means that you have to make a seat reservation before boarding the train.

Will I be able to reserve a seat when I just get there?

Yes, unless the train is booked out, which does not usually happen.

From Tokyo to Kyoto, is a green car on the Shinkansen spacious enoguh for my luggage?

A green car seat has enough space for you to actually put your suitcase in front of you in case that it does not fit onto the overhead shelves or behind the last row of seats:

Depending on the size of the suitcase, it might even fit in front of you on an ordinary seat:
http://www.japan-guide.com/e/e2274.html

From Kyoto to Osaka, how much would a Taxi cost?

Depends from where in Kyoto to where in Osaka. Probably around 10,000 yen. I recommend to travel by train:
http://www.japan-guide.com/e/e4006.html?aFROM=2157_2158

What is the best option to get to Kansai Airport, from Osaka?(with my luggage)

Again, this depends on where in Osaka you will start your trip. Here are general options:
http://www.japan-guide.com/e/e2033.html

What I usually say is if your luggage is the size where it can fit into the overhead luggage bin of an airplane then you are generally fine getting onto the overhead space of trains.

The green car doesn't necessarily have any extra overhead space, that space is the same as you find in ordinary cars of the Shinkansen.

The Narita Express has additional luggage space at the end of each car for large luggage.

I took the NEX from Narita to Shinjuku and arrived in Shinujuku stationa around 10am+. I had no problems navigating the station at all though it was crowded. People generally avoid you when they see you dragging 2 big luggages. And the man at the train gates were really kind, he helped us to carry some of our luggages pass the train station barriers.

I travelled each route you have specified by shinkansen (regular, not green car & excluding Nozomi as I was on a pass) Both my partner and I were carrying 75L backpacks and found we could place them between our legs and the seat in front of us. It was a little cramped, but very much possible.

You seem to be intimidated by the transport system- I was also before I arrived. Let me say, as a gaigin with very little Japanese ability, I had no trouble navigating any of the JR system, many announcements are made in English, and much of the signage is also english- this includes departures and arrivals)