Should I download Lion first then Mountain Lion?

It isn't; it costs more and increases the chance of experiencing trouble while upgrading.

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You can upgrade directly to Mountain Lion from Snow Leopard. Furthermore, you will save a lot of time and money, and you may get a better performance. There's no reason to upgrade to Lion first.

Open the App Store and purchase OS X Mountain Lion. While it's being downloaded, make a backup of your files and check that your applications are compatible > http://www.roaringapps.com

Upgrade directly to Mountain Lion from Snow Leopard:

Upgrading to Mountain Lion

To upgrade to Mountain Lion you must have Snow Leopard 10.6.8 or Lion installed. Purchase and download Mountain Lion from the App Store. Sign in using your Apple ID. Mountain Lion is $19.99 plus tax. The file is quite large, over 4 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.

OS X Mountain Lion - System Requirements

Macs that can be upgraded to OS X Mountain Lion

1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later

2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model Identifier 5,1 or later

3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later

4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later

5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later

6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later

7. Xserve (Early 2009) - Model Identifier 3,1 or later

To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.

Are my applications compatible?

See App Compatibility Table - RoaringApps.

For a complete How-To introduction from Apple see Upgrade to OS X Mountain Lion.