Can I extend my contract before I ship out to basic training?

I'm certain the MEPS would allow you to renegotiate for a longer term because it would benefit the Army. When you ship out you will return to the MEPS and sign another contract stating your term of service. At that point you can request for a longer term.
I would caution you against that. When you are at your first duty station, you will be eligible to reenlist for a choice of duty station once you are 15 months from your ETS. Or if you like the location you can reenlist to stabilize where you're at. Either way, reenlisting will bring options to you that won't be available if you enlist for 6 years.

No don’t do it I’d definitely feel it out first and do 3 years first.

Unless your going to get a massive bonus for doing more than 3, just stick with 3. Your reenlistment window will open up sooner giving you a vast amount of choices such as choice of duty station, special training, bonus money etc.

Every time you are able to reenlist you have the opportunity to qualify for a reenlistment bonus.
If you only signed for three years you'll be eligible to re-enlist in two.
Keep in mind, the Army wants you to stay in! The more training and experience you receive the more valuable you are.
Also, although you signed a three year contract it's actually three by five years. That'll be three years active/reserve and five years reserve/IRR. Until you complete eight years you won't truly be released from the military.

I suppose you could bring it up at MEPs if everything isn't finalized, including taking the oath and receiving the infamous bend over exam. However, note that combat engineers have a tough and physically demanding job... with lots of cool training that sound cool... but once the cool factor wears off then what... Also, the army is the army and it may not be exactly as you pictured or hoped for. Additionally, once you commit to a unit it's not going be easy to just transfer to a new unit or reclass to another MOS on a whim. You might also get stuck in a unit with people you can't stand and you'll have to tolerate it. You might also have unforeseen family emergencies which will conflict with your duty/responsibilities as a soldier since the Army usually comes first. Can you honestly say your life will remain as it is now without any family issues for the next 6 years?

You will need to speak to your recruiter. There are varying lengths based on your component, AD, Guard or Reserve.

Join the Marines. It's not too late.

Wait until you are close to your ETS date. Between 12-15 months out, assuming you aren't admin flagged, you will be approached by whoever is the unit Retention NCO. At that point you will have great bargaining power. As long as you are a good soldier, the Army will WANT to keep you. That is when they will offer incentive for you to stay enlisted. It costs the Army less money to keep a soldier than to train a new one. Use that to your advantage to get cash bonuses, duty station changes, MOS changes, rank, etc.

You probably can, but honestly why? The reason I ask this and question is that most, when they go in are hyped up. They think they know what it will be like and is like but they don't. They really don't. Some recruiters lie. Or at the very least omit certain basic truths.

You need to wait. Go to basic, then go to AIT and on to your first duty station. You will get a good idea of military life by then. It might not be something that is really for you or that you really want. If it is, then at the end of your initial enlistment you can always re-enlist. That is one of those things that never really goes away. And there is no rush. You do not earn any more money for enlisting longer. You do not advance faster and you are not viewed more favorably for enlisting in a longer term.