I have counted 20 episodes in which the TNG Enterprise travels to places where there has been no presence from an existing manned Federation ship or colony and no presence from an existing superpower that's known to have shared survey information with the Federation (e.g. Romulans, Cardassians, Klingons, etc). I have also excluded episodes where the Enterprise is trying to find a lost ship or a missing colony.
I have indicated the place, the purpose of the exploration and whether any first contact was made with an alien life-form.
Season 1:
- Where No One Has Gone Before* (Another galaxy - By accident)
- Justice (Strnad System - Planetary survey - First contact mission)
- When The Bough Breaks (Epsilon Mynos System - Planetary survey - First contact mission)
- Home Soil (Pleiades Cluster - Survey mission)
Season 2:
- Where Silence Has Lease* (Weird space hole thingy - By accident)
- Pen Pals (Selcundi Drema System - Planetary survey x 5 - Partial first contact)
- Q Who* (J two five - Planetary survey - Attempted first contact mission)
- Samaritan Snare (Epsilon Nine Sector - Stellar survey)
Season 3:
- The Bonding - (Koinonian System - Archaeological survey - First contact mission)
- The Offspring (Selebi Asteroid Belt - Asteroidal survey)
- Tin Man (Beta Stromgren system - Stellar survey - First contact mission)
- Transfigurations (Zeta Gelis Cluster - Stellar survey/planetary survey - First contact mission)
Season 4
- Clues (Ngame Cloud - Planetary survey - Partial first contact mission)
- Nth Degree* (Centre of the galaxy - Planetary survey - First contact mission)
- In Theory (Mar Oscura - Nebular survey)
Season 5
- The Game (Phoenix Cluster - Stellar survey)
- Imaginary Friend (FGC Four Seven system - Nebular survey)
- The Inner Light* (Kataan system - Planetary survey - Accidental first contact)
Season 6
- Schisms (Amargosa Diaspora - Nebular survey - Partial first contact)
- The Chase (Volterra nebula - Nebular survey - Partial first contact)
Season 7
There are no significant explorations in Season 7.
* indicates episodes where there was a distinct lack of boldness, where the Enterprise traveled timidly or under duress.
Into Darkness, that's the Starfleet Captain's Oath (citation needed? unsure). – AStopher Mar 19 '17 at 12:17