Before all else, let me say this: Since Gen 6 or Reborn's E13, TMs are no longer passed on through breeding, but both parents can pass on Egg moves now. Until Gen 5 or Reborn's E12, you could pass on TMs, but only the father could pass on moves, TM or Egg, so you may have to switch episode, depending on what you need/want to do.
Aight, so let's start with the very basics of breeding. Egg Groups. If you check any Pokemon on Bulbapedia, you can see an Egg Group tab, and 1 or 2 listed groups, like so:
-Now, only Pokemon that share an Egg Group can breed and produce an egg. Noibat and Zubat are both members of the Flying Egg Group, so they can breed. Zubat and Muk are different groups, so they can't.
If the participants have 2 groups each, only one has to match. Farfetch'd is in the Flying and Field groups, Wingull is in the Flying and Water 1 groups, but since both are in the Flying group, they can breed.
-To breed, you need a male and a female "parent". Whatever hatches is of the same "species" as the female parent. There are some exceptions, however. Genderless Pokemon and those in the Undiscovered Egg Group cannot breed.
-There is only one Genderless Pokemon that can breed with Gender-having Pokemon: Ditto. Whatever breeds with Ditto counts as the mother, regardless of actual gender. As such, this is the only circ*mstance in which Genderless Pokemon such as Metagross or Magnezone can breed, but moves cannot be passed down when breeding with Ditto.
-Keep in mind, although the gender of the new Egg/Pokemon is randomized, keep in mind that there's species like Combee (7:1 male-female ratio) in which case it is still random what gender you get, but you're more likely to get the egg of a male one. Unless you're breeding a genderless Pokemon with Ditto, in which case the result is genderless still.
-Now, the actual point, or points, of breeding. There's 3 purposes of doing this: modified IVs, "changing" abilities and passing on moves.
1)IVs can be seen in Reborn when you press Z/Space/Enter on the stats screen of a Pokemon's summary, the numbers on the right side. Each stat has one, they range from 0 to 31 and, basically, at level 100, they give you 0-31 stat(s) point extra, the lower the level, the less benefit you get from IVs, so, say, 26 IVs in Attack give you 13 Attack points at level 50.
-In comparison, the difference between a neutral and a +nature for a stat is ~25 point difference at level 100.
-IVs can be passed on through breeding. For 3 IVs, a random parent is chosen and then a random stat is chosen, the egg will have the same IVs in that stat as the parent had in that stat, the other 3 are randomly generated.
-But if one of the parents holds a Destiny Knot, of the 12 IVs in total both parents have, 5 stats are passed down and the 6th one is randomly generated.
-Also, if the parent is holding a power item, the IV of the stat said item affects will definitely be passed on. So if the parent has a Power Anklet equipped while it's in the Day-Care, the new Pokemon is guaranteed to have the same Speed stat as said parent.
2)The second point is not as important in Reborn, but in regular games, if the mother has it's hidden ability, which is rare/hard to get in other games, the new pokemon has a 50% chance of having it's hidden ability.
-In Reborn, Hidden Abilities are as frequent as normal ones, so that's not as important in Reborn.
3)Now, the most important part, and probably why you're even here-passing on moves.
-The most important moves that can/could be passed on pre-E13 are TM moves. Whether a TM move can be passed on depends on whether the new Pokemon's first evolutionary step can learn that TM.
-By first evo step, I mean that even though Crobat learns X-Scissor by TM, Zubat doesn't, so it cannot get X-Scissor by breeding. Zubat can learn U-Turn through TM, so getting it through breeding is an option.
-Next, Egg moves are non-TM moves that can still be passed on through breeding. Those moves are listed on a Pokemon's page on most sites that have info about them, but Bulbapedia is the most accurate and also lists exactly who gets what from who, so just use that. They're listed below the moves learnt through TM, with a "By breeding" header.
-If you'd check Noivern's page, the egg moves listed Outrage, Snatch, Switcheroo and Tailwind. Check the symbols at the bottom of that box to see what they mean.
-Egg moves are the same for all evolutionary stages of a species, unless the first stage is a baby Pokemon (which we'll address later), Gastly and Gengar have the same Egg moves.
-At the start of this "section", I said the most important moves, because you could also pass on a move the "child" learns by leveling up if both parents have learned that move, but just waiting and leveling up the new Pokemon is less of a hassle.
-Let's look at Gengar now. Through breeding, even without TMs Gengar can easily learn both Sludge Bomb and Thunderbolt pre-E13. First, for moves we want, we see whether the lowest evolutionary point can learn the desired moves. Gastly learns T-bolt and S.bomb, so we're good to go.
-Next, we need Pokemon which learn Sludge Bomb and Thunderbolt through level-up and are in the same Egg Group as Gastly, which is Amorphous. So we need Gastly/Haunter/Gengar, Stunfisk and Grimer/Muk. They are all in the Amorphous Egg Group, so they can all breed with each other, Grimer learns Sludge Bomb by leveling up and Stunfisk learns it through level-up.
-First, we take a male Stunfisk and level it up to level 45 at which points it learns Thunderbolt through level-up.
-Then we breed it with a female Grimer/Muk. The result will be a Grimer Egg with Thunderbolt.
-You need a male Grimer, since we want a Gastly Egg at the end, so the Gastly has to be female and it has to breed with a male Grimer/Muk.
-Then, level that male Grimer/Muk until it learns Sludge Bomb and breed that male Grimer with Thunder Bolt and Sludge Bomb with a female Gastly.
-You'll get the egg of a random gender Gastly with Thunder Bolt and Sludge Bomb.
-You can also breed "through" Egg Groups, fortunately. As I said, 2 Pokemon have to have at least one shared egg group. Only those 2 current parents, not all participants of a breeding chain.
-Let's say we want an Archeops with Earthquake. Archen learns Earthquake through TM, so we're good to go.
-It's in the Flying and (curiously) Water 3 egg groups. No member of either group learns EQ through Leveling up. However, some members of those groups learn EQ through TM as well, so we see which egg groups those are in and if someone in those groups learn EQ by leveling up. Repeat if necessary.
-Now, in the Water 3 Group, there is Corsola. Corsola is also in the Water 1 group, in which there is Wooper, who learns EQ through leveling up.
-So, we have a male Wooper learn EQ by leveling up and breed that with a female Corsola. Both are members of the Water 1 group, so they can breed. Next, we breed the new male (hopefully, get a new Egg if not) Corsola with Earthquake with a female Archen/Acrheops, to get the Egg of an Archen with Earthquake.
-The nature of the new Egg is usually random, but if you give either parent an Everstone to hold during breeding, the Egg will have the same Nature as that parent.
-Also, should you look at the moves a first evo-stage Pokemon learns by leveling up on Bulbapedia, you'll see some moves are learned at level "Start", which means that a hatched Pokemon will have those. However, if the number of Start moves plus the number of passed-on moves is higher than 4, breeding moves have priority.
-Should you breed a male Spiritomb with Shadow Ball, Memento and Nasty Plot (Memento and NP are Yamask's Egg moves, SB is a TM move Yamask can learn) with a female Yamask, which "learn" Astonish and Protect at Start, you'll get a Yamask Egg with Astonish, Memento, Nasty Plot and Shadow Ball.
-Since Gen 6, both parents can pass on Egg moves, but no TMs. It gives entirely new options. Let's take a look at Azumarill. 2 interesting Egg Moves it has are Belly Drum and Aqua Jet. Now, in Gen 5, it is not possible to get a Pokemon to have both of those and breed it with a female Azumarill to pass both moves on, you'd have to settle for one. But, since gen 6, you can breed a male Poliwhirl with Belly Drum with a female Azurill, get a female Azurill with Belly Drum and breed that with a male (based on what's in Reborn) Floatzel/Kabutops/Empoleon/Carracosta/Golduck with Aqua Jet. The final result will be an Azurill egg with Aqua Jet and Belly Drum.
-Another thing worth mentioning is pre-E13 Smearge. It's signature move Sketch lets is copy any move. In which case, if Azurill was in the same Egg Group as Smeargle, the Field Group, you could have male Smeargle learn both Aqua Jet and Belly Drum like so and breed it with a female Azurill/Marill/Azumarill to get both moves in one go.
Here's a link to a wonderful guide to Smeargle by Gutten_Tag.
-Oh, and one last thing - Baby Pokemon. Mantyke, Riolu, Munchlax, Happiny, Mime Jr., Bonsly, Chingling, Budew, Wynaut, Magby, Elekid, Smoochum, Tyrogue, Togepi, Igglybuff, Cleffa and Pichu (although Wynaut doesn't get any TM moves/Egg moves anyways) cannot breed at all. To do any breeding, you have to evolve them at least once to do anything. However, in terms of TM breeding, you still have to check what TMs they learn in order to breed 'em onto a Pokemon.
