It's long press the physical button. Only HTC phones have a multitask button.
Actually, you can set up various ways to do this regardless of phone. there is a free app called Swipepad that allows you do create any virtual task like opening recent apps, or opening a specific app, or going to the homescreen. no rooting required.
Nova and Apex launcher allow for physical home button manipulation via long press.
rooting your phone and loading a custom ROM like AOKP will allow for the creation of any vitual nav button and long press you want. This includes "recent" or "multitask''.
(i am sure you know all this. my info is more intended for the OP.
As for internet, use Chrome. Once you have a Google account everything you do including Chromes bookmarks are synced aautomatically and will always be available regardless if you are on a pc, mac, tablet, phone, public computer. Logging into Google will auto populate Chrome. Samme with any docs you have, aanything you put into your Calendar, GMail, etc
One option within Google for Chrome is to never record your history. And it works. I use this feature, but you need to log into your Google account to do this, so your history will never be recorded regardless of any device you use. Cache can be wiped via Chromes settings.
By the way, on the Note 2 you can open two apps at once via split screen. This is a feature coming to the S3 as well.
Basically, anything you can imagine can be done. I have an S3 and my wife has the Note2. It is an awesome phone. Once you use it and become accustomed to Android and the screen size, the iphone will look like a toy phone.
