How can I take everything from the current SSD and then upload it to the new SSD - effectively giving me the same machine without installing OSX all over again?
I upgraded my SSD some time ago in a similar configuration (128 GB->256 GB, MacBook Pro 15-inch early 2011).
Below is the procedure that worked for me, but note that for this to work your Mac must support OS X Internet Recovery. According to KB article "OS X: About OS X Recovery" (http://support.apple.com/kb/HT4718):
Mac models introduced after public availability of OS X Lion include
the ability to start up directly from an Internet-based version of the
OS X Recovery system.
which means that models manufactured before July 20, 2011 need a firmware update.
In your case you must have MacBook Pro (13-inch, Mid 2010) EFI Firmware Update 2.5 or later installed, and that's probably the case, because most firmware updates are available as regular updates in Software Update/App Store.
To find out whether it is installed: Go to Apple menu > About This Mac, press System Report...:
and check if Boot ROM/SMC Version Information is
MB71.0039.B0E or greater. If that's the case, you're good to go. Otherwise install it.
These are the steps:
Back up old SSD with Time Machine by selecting "Back Up Now" in menu bar:
Shut down Mac and replace SSD.
Start Mac while holding down ⌘-R and choose "Restore From Time Machine Backup":
Choose new SSD drive as target, wait until restore is completed and reboot.
This will let you keep everything.