I did my first voyage of discovery in a Type-6 and found my first completely undiscovered system about 500ly form Sol. This was about 3 months ago. I headed up towards to Polaris and then zig-zagged towards the most uninteresting looking bit of sky.
Nutter's explorer's guide really helped get me going on the understanding and exploration and discovery.
https://forums.frontier.co.uk/showthread.php?t=81653
Some basics tips on discovery:
Make sure you have your discovery scanner bound to a trigger in the fire groups. When you enter a new system hold the trigger and wait for the little blue bar to fill and for it to go ping. This will populate the system map. Make sure you are not too close to the star before you open the system map so that you do not overheat.
The system map may not show all the objects in the system. Only those within the range of your discovery scanner will be shown.
If you hover over objects in the system map you can see whether somebody has already discovered them. Those objects already discovered will have DISCOVERED BY and the Commanders name in the information pop up. If you see no name then you can be the first person to discover it!
To "discover" a body you need to target the object, just like you would another ship. When locked on the target a yellow circle will appear on your HUD. Keep pointing at the object for the scan to work, this takes between 35-15 seconds depending on how close you are. Every object has a minimum distance in order for the scan to work. The smaller the object the closer you will have to be for the scan to work.
A detailed surface scanner is a great investment for any explorer. It will provide extra information on the body and at least double the payout for the discovery. However it is not necessary to have one in or to discover a body if you can't afford one yet.
It is worth scanning objects even if they have already be discovered by another Commander. Stars, gas giants, earth like worlds and high metal content planets all pay well. Check Nutter's guide for the full list of payouts. Being the first discoverer of an object gets you a 50% payout bonus, your name on the system map and a nice fuzzy feeling.
Be aware that your discoveries only register when you sell them to Universal Cartographics so make sure you make it back to civilisation in one piece!
I suggest you post here how much money (total assets) you have and we can help make suggestions on how to build an explorer ship for your budget. I you have Horizons I strongly suggest you take an SRV. It's a special feeling to drive on the surface of a planet in a system where no other commander has been before. You can also get some great screenshots to remember it by.
Fly safe Commander!