How should I move data from old hard drive?

When you get the new HDD, put it into that enclosure for 2.5" S-ATA HDDs and then format it via Disk Utility* properly**.
Then use CarbonCopyCloner or SuperDuper to make a 1:1 copy of the internal HDD to the new external HDD, then swap the drives and use the old HDD for whatever purposes you intent to.

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simsaladimbamba said:

When you get the new HDD, put it into that enclosure for 2.5" S-ATA HDDs and then format it via Disk Utility* properly**.
Then use CarbonCopyCloner or SuperDuper to make a 1:1 copy of the internal HDD to the new external HDD, then swap the drives and use the old HDD for whatever purposes you intent to.

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Links to guides on how to use Disk Utility, the application Mac OS X provides for managing internal and external HDD/SSDs and its formats.

Guide on how to use Disk Utility from the restore / installation DVD.

Format Your Hard Drive Using Disk Utility
Partition Your Hard Drive With Disk Utility

Using Disk Utility - Formatting, Partitioning, Verifying, and Repairing Disks

DiskWiping – One Pass is Enough​

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How to format a new HDD/SSD to install Mac OS X or make a bootable copy onto​

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Maybe have a look at Advanced Search to find similar threads, as this question comes up a lot:

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Click to expand...

Is it possible to just insert it into my mac? Will it start downloading Mountain Lion from the internet and then will I be able to restore from my time machine backup?

I do now have a HDD enclosure and I will be upgrading to a samsung 830 ssd basic kit.

EarthNeutron said:

Is it possible to just insert it into my mac? Will it start downloading Mountain Lion from the internet and then will I be able to restore from my time machine backup?

I do now have a HDD enclosure and I will be upgrading to a samsung 830 ssd basic kit.

Click to expand...

You have to format the HDD at least, and as far as I know, currently you can only download Mountain Lion from the Mac App Store and not via Internet Recovery.
Why not download ML on your current Mac with your current HDD, make a bootable installer (many, many guides available on how to do this with a 4 or 8 GB USB flash memory thumbdrive), insert the new SSD, make a clean install of ML (which includes formatting the SSD correctly, again, many guides available for this and for Lion (same procedure))and then either use Setup or Migration Assistant to import the data from the Time Machine backup.