How to increase territory through diplomacy?

You may engage other nations' leaders in diplomatic negotiations, (un)fortunately there is only one proper dialogue chain for each nation.

Out of 26, I know of 20 nations that are eligible to be annexed diplomatically. The rest (Narbonensis, Sarmatia, Dacia, Scythia, Parthia, Carthage) are immune to diplomacy - the only way to control those regions is to conquer them. (Scythia is an exception - diplomacy is possible in theory, but they refuse whatever tribute you demand... probably a bug).

Some things I've tested personally, some I've gathered from various sites. If there's any discrepancy, I'll correct it as soon as possible.

General tips

• Keep your subjects happy (especially Rome). Diplomacy is near impossible with Rebellious Romans (that's why usually you don't succeed at diplomacy for the first few years - Rome's poor happiness level is to blame)
• Don't plunder. Plundering renders every nation aggressive towards you for a couple of turns and diplomacy will fail.
• Rank up. Higher rank means bigger armies. You advance mainly by winning combats, racing and gladiator fights can also improve your rank.
• Winning races, hosting gladiator fights, winning battles and gaining territories help with achieving higher rank.
• All the above tips can be condensed into one point: There's a "Roman Reputation" counter which helps with diplomacy, where won battles/fights/races and gained territories increase reputation, and losing battles/fights/races and territories decrease it. Unfortunately, there's no visible display of this reputation. It's closely connected to your rank though. However, once I tried to ally Alpes but they didn't want to, and after gladiator fight they accepted. I didn't get new rank in the following turn, the only difference was the fight. Also, several sites claim that you need at least Tribune to get Dalmatia, but I managed to ally them once while being Centurion. I did get Tribune in the following turn though.
• Usually, you need bigger or equal army to succeed in negotiations (Some nations will laugh at you if your army is too small). However, there are 4 nations where you actually need to bring smaller army (namely Sardinia, Asia, Pontus, Syria).
• Generally, when they're asking you for tribute level, you've won the negotiations, but...
• Choose "Low Tribute", just to be on the safe side - some nations will accept higher, but you can change it manually anyway.
• If you wish to ally as many nations as possible, start with attacking Narbonensis, then wait until you can ally Sicilia/Dalmatia/Alpes. After that, it's really up to you.
• If you lose Rome, the game might start behaving weirdly, like graphic glitches, diplomatic options doing other things, endless diplomatic loops or freezing. Don't lose it :)

Countries:

• Sicilia - 3600 men
  • Diplomatic - Aggressive - Offer Alliance
• Dalmatia - 4600 men
  • Aggressive - Offer Alliance
• Alpes - 3000 men
  • Aggressive - Friendly - Diplomatic - Offer Alliance
• Germania - 8000 men
  • Aggressive - Friendly - Diplomatic - Offer Alliance
• Gaul - 7000 men
  • Aggressive - Friendly - Diplomatic - Offer Alliance
• Hispania - 5340 men
  • Diplomatic - Aggressive - Offer Alliance
• Thracia - 3300 men
  • Diplomatic - Diplomatic - Offer Alliance
• Macedonia - 4600 men
  • Diplomatic - Diplomatic - Offer Alliance
• Armenia - 5000 men
  • Diplomatic - Diplomatic - Offer Alliance
• Mesopotamia - 3900 men
  • Diplomatic - Diplomatic - Offer Alliance
• Arabia - 3900 men
  • Diplomatic - Aggressive - Offer Alliance
• Aegyptus - 5120 men
  • Friendly - Friendly - Offer Alliance
• Mauretania - 3340 men
  • Diplomatic - Aggressive - Offer Alliance
• Britannia - 6000 men
  • Aggressive - Offer Alliance

Sardinia, Asia, Pontus and Syria - SMALLER army required.

• Sardinia - 2300 men
  • Friendly - Friendly - Offer Alliance
• Asia - 3640 men
  • Friendly - Friendly - Offer Alliance
• Pontus - 4320 men
  • Friendly - Friendly - Offer Alliance
• Syria - 3340 men
  • Friendly - Friendly - Offer Alliance

Cilicia and Cyrenaica will ally you under certain circumstances. One idea is that you need to plunder a province first. Another - you need to earn 100 talents/year in tribute. Third, and most probable, is that you need to have rebellion or lose a province to an enemy army. When I figure out exactly which one, I will update this answer.

• Cilicia - 3600 men
  • Friendly - Offer Alliance
• Cyrenaica - 2940 men
  • Friendly - Friendly - Offer Alliance

Many people have asked how to get an alliance with Cyrenaica. The path for doing this is somewhat complicated. Here is a Youtube comment that describes how to do it:

How to ally with Cyrenaica:

Basically, you need to lose some territories and then regain them in order to make them ally with you. This is what works for me so far:

1. By "losing a territory" I mean losing it to a marauding army. Losing it due to rebellions does not seem to work for me.
2. The territories must not be ones that are wealthy, or ones that have Fierce soldiers. If you lose one of these territories, Cyrenaica will be unwilling to ally with you, even if you regain it immediately. Usually, I try to lose and regain the mediocre territories close to Italia (such as Alpes, Dalmatia, and Narbonensis), which will not make Cyrenaica refuse me.
3. After you lose each territory, you must move in an army and regain it immediately.
4. You must lose (and regain) 2-3 territories in this fashion.
5. The interval between territory losses must not be too long (I'd say 2 turns to be on the safe side).
6. You must march into Cyrenaica very shortly after you have lost and regained your second or third territory.

The command sequence with Cyrenaica is Friendly - Friendly - Offer Alliance - Low. As mentioned in another comment, if the response to your first command is something like "We welcome our fellow brigand. We are as brothers" this means you are on the right path.

Additional requirements:

1. You must march in with at least a Cavalry-sized army.
2. You need some rank (worked for me with General rank).
3. You need at least some victories against high-quality opponents (such as Carthago, Macedonia, or Parthia) before Cyrenaica (or any other country, for that matter) is willing to ally with you.