What is the best opening after e4 e5 nf3 nc6 bc4 bc5?

That is what I said dude

TINSTAABO
"There is no such thing as a 'best opening.' Each player should choose an opening that attracts him. Some players are looking for a gambit as White, others for Black gambits. Many players that are starting out (or have bad memories) want to avoid mainstream systems, others want dynamic openings, and others want calm positional pathways. It’s all about personal taste and personal need.
For example, if you feel you’re poor at tactics you can choose a quiet positional opening (trying to hide from your weakness and just play chess), or seek more dynamic openings that engender lots of tactics and sacrifices (this might lead to more losses but, over time, will improve your tactical skills and make you stronger)." - IM Jeremy Silman (January 28, 2016)
https://www.chess.com/article/view/opening-questions-and-a-dream-mate

There's a lot of lazy thinking here. 

If there isn't a clear win or a way to get a clear advantage as White, the best play is to keep the position full of life. 

In that sense, there is a best line of play for White after 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5.

That is, White should play c3 and d3 with the idea of deciding whether to play d4 soon, castle K-side or Q-side. This flexibility is clearly White's best course of action, and is played quite often by top players against other top players. 

Today, Kramnik defeated Matlakov, rated 2718, by playing just that sort of flexible plan. In this case, he chose to castle Q-side and played a slightly unusual plan. It worked for him because he knew how to play the resulting position and was more comfortable than his opponent. 

In general, the idea that any reasonable opening is good enough to get a playable game is true. But in the specific case of the Italian game, White does best to play flexibly.

Does that mean that other opening choices such as the Evans are bad? No. But it's less flexible and less likely to produce a result against a well prepared and strong opponent.

What's faster?  Going through MCO or going through the Opening Explorer of 365chess?  What's an easier way to learn the various lines of an opening?

If you really want to learn how to play an opening, select ~50 games played by experts in the opening and play through them. Take about 10 minutes per game, asking questions as you go along. This first iteration is just to get a feel for the opening. 

Take a look at the questions you asked and see if you can answer them. Pay attention to the move orders used, and the dates of the games. Did the move order evolve over time, or has it been consistent? If the move order is more or less the same for ~20 or 30 years, there's probably a good reason for it. Experiment with different move orders against an engine and see if there's a flaw to other move orders. 

Ask more questions, take clear notes. 

When you've done that, take a look at some annotated games and see what you missed. 

At the very end of all that, go to an opening book and see what the variations are. You probably already know most of it, but that's the point! You will have learned more about the opening than is usually covered in an opening book. And the lines will make sense to you!

Caveat: This isn't a fast way to learn an opening, but you will learn the opening.

Thanks Smyslov Fan.  Just thinking out loud.  50 games times 10 minutes per equals 500 min.  Divide by 60 equals 4 hours and 20 minutes. 

I would think then I would learn it from a computer game database, clicking as I go to keep within 10 min per game, yes?

Does a Class player learn best via screen with time savings or setting up the board and taking longer with all the resetting of the physical board and pieces?

"Actually the modern way to handle the d3 lines is like (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5) 4.0-0 first, as white does not have a meaningful way to get something after 4.c3 Nf6 5.d3 0-0 6.0-0 d5!?" - IM pfren (December 13, 2017)

Pfren might have been right, but Kramnik didn't play 6.0-0.

Added: Also, Pfren is mostly interested in correspondence chess. I'm mostly interested in over-the-board chess, which is more practical and often less theoretical.

@SeniorPatzer, the method of learning, either via a screen or tactile learning by moving the pieces, depends very much on the individual. I generally learn better when moving the pieces. I go slower and remember the moves better. But not everyone is the same. You could just use the screen and make sure you spend the time to memorize the moves. Make sure you visualize the moves clearly before continuing. 

Use the method of learning that works best for you. 

"FIDE IM, FIDE Trainer Available for complete INDIVIDUAL courses to abitious advanced young players, aiming at international titles."

Did Kasparov ever play 4 b4 ?

Yes. Kasparov played the Evans Gambit against Anand in 1995 and won, in their last tournament game against each other before their world championship match.

Shut the frickin up will you people. e4 e5 SUCKS anyway

Dude!

d3 

c3

yeah, actually b4 is good