Good communication between ADC and support is something that everyone recognizes as being important---during laning, it's arguably the most important communication channel in the entire team. Here are some tips on what you should say to each other. I will assume for the most part that you have voicechat and then at the end briefly discuss what you should take the time to type in the event that you don't. The four types of things you need to talk about are what to do; when and how to do it; whether you're able to do it; and the micro of fighting as two separate champions.
Note that a lof of this advice applies to any interaction within a team, but priorities shift when you're talking to someone who's not in the same lane as you.
Before the game even starts, discuss briefly how you want the lane to go. For example, if you're playing Leona-Corki, you might say "We're going to take doubles and then soft push the lane at level 1, and then the instant we hit two, Leona is going to E to Sona and we're going to focus her down really quickly since Vayne has cleanse. We need to make sure we get a kill before they hit 2, and then we'll try to snowball of that. If we don't get a kill at level two then we'll ask the jungler to help and then wait for a gank to go really aggressive unless we are able to heal enough after the first fight that we can continue going allin." If, at level 2, you don't get the kill, you should briefly discuss what went wrong so that you can avoid making the same mistake later on IN THE SAME GAME. Don't worry about future games right now; that's what replays are for. Then remind each other of the initial plan; if it changed, say what you're planning on doing next, and if not, then just continue.
Don't dwell on this sort of macro-communication too much. You need to make sure that both of you understand your goals, but you don't want to distract yourselves from laning, since thinking about long-term plans can be very distracting not only from the mechanics of laning but also from more immediate things that you need to say out loud. When you're first starting to play together as a botlane, you'll need to dedicate more time to this sort of discussion than you will once you've played 50 games together. Once you're very used to each other as players, this should be at most 10% of your communication.
Once you both understand how the lane will work, you need to make it actually happen. Talk about your short-term plans. Things you might say here are, "I'm going to flash-pulverize Ashe in 3, 2, 1, go" or "After I get these two creeps, I will walk forward. Walk up with me because we can kill them or push them out of lane now" or "When this pushes to tower, we can dive. Tank for me and we focus Lux, just don't get baited by her shield" or "I need to go ward tribush. Can you walk with me so that I don't get caught out as I round the corner?"
Discussing short-term plans like those should probably be the majority of your communication. If you have the rest of your team in voicechat, try not to hog it too much, as the jungler will be needing to coordinate with everyone at the same time, but definitely say everything that you need to. As you start to play together as a botlane more and more, you will need to say things like this out loud slightly less, but you'll never stop talking about short-term plans, even if you eventually start saying things like, "Go?" "Okay, 3, 2, 1, go" because you both know what that means. In a premade team it is a good idea to abbreviate your sentences as much as possible without losing any information so that you're not interfering with the rest of your team's communication too much.
The next thing you need to talk about is cooldowns. It's very easy to fall into the trap of not communicating your cooldowns, but it's something you really need to keep on top of. Roughly every thirty seconds, you should state the cooldowns on your ults and summoners, and every time you make a short-term plan, you should announce what spells you have (if relevant). For example, you might say, "Exhaust in 10, ult in 30, flash in two minutes...exhaust up, ult in 20, no flash...10 seconds on ult, I have exhaust, dive as soon as it comes up...ult in 3, 2, 1, go." Or you might say, "Amumu, can you come bot? I'll have ult up when you get here, both my summoners are already up." Not only is it a good idea to state your long cooldowns out loud so that your lane partner knows what you have up, but it also ensures that you won't miss the fact that one of your summoners came up and then not use it in a fight because you thought you didn't have it.
For certain champions, you want to keep track of more than just your ult and summoners. On Soraka, I try to tell my carry the cooldown on my heal every five seconds. If he's low on mana too, I'll track the cooldown of my E out loud also. If we're trading very frequently, I'll track skills like Sona Q or Janna E every two to three seconds. Carries should track long-cooldown nukes like Graves Q or Ashe W and their escapes, like Ezreal E and Tristana W. In certain cases, like Kog-Nunu, the carry should also announce their steroid's cooldown.
Announcing cooldowns can take up 100% of your communication in theory, as you can always say what you have ready. This is obviously not practical, but if there is a lull in the conversation, you should announce what skills you have up and the cooldowns on the remaining skills.
The other part of calling out cooldowns is calling out mana costs. Maybe Sona has her ult up, but she only has mana for ult and one spell, which is not enough to fight because she's only at one stack of power chord. She needs to say, "I have ult but not enough mana to fight," so that her carry knows that she can disengage with her ult if need be but that they shouldn't go aggro until Sona regens enough mana for a couple more spells. It can also be useful to call out how many potions you have left---"I have two HP pots left, and their Graves doesn't have any pots left, so we can trade pretty aggressively since I can outsustain him by a lot and they don't have kill potential" or "I'm out of HP pots and only have one Dblade. Graves has two pots left; let's try to avoid trading until our jungler gets here" are two entirely different situations, even though both of you may be at full HP and mana at that time with all of your cooldowns up.
The last thing to communicate is probably the hardest: micro. Saying things like "flash to me, my shield is up!" or not can be the difference between life and death, but it can be very difficult to concentrate on your own micro at the same time. A good way to practice micro communication is to say everything that you're doing out loud when you're playing by yourself or without voicechat.
No one of these four types of communication is the most important, and you will frequently not distinguish between them very much in practice. You need to state cooldowns in the middle of teamfights, and you should plan out how you will execute the plays you are planning to attempt before actually starting them. However, it can be helpful to categorize your communication into these separate types to make sure that you're not missing anything, especially when you're just starting to play as a duo bot.
Frequently, a duo bot will have a playcaller, even if the team's overall playcaller is the midlane or jungler. The playcaller will be the one to say when you're going to aggress, when you need to play slightly more passive, and when you go back to base. This means that the person who's not the playcaller needs to be even more on top of stating cooldowns and mana costs so that the playcaller is working with complete information. Not having a playcaller is ok, but you should both know before the game starts who you will defer to in case you disagree. This last part is extremely important---it's usually worse to do different things than it is to both do the incorrect thing together.
When do you ignore the person you agreed to defer to? The short answer is never; the longer answer is when ONLY YOU HAVE VITAL INFORMATION YOU ARE 100% SURE THEY DO NOT HAVE (for example, you are Soraka and you just went to ward river; on your way you healed the jungler, and your carry doesn't know that your W doesn't come up for another 5 seconds). And in this case, you already made a mistake because you should have communicated this information earlier. The correct course of action to take here, is to tell them the information that they're missing IMMEDIATELY. Corki is about to valk in? Say, "I don't have heal" the instant he says he's going to do that, but be ready to flash for an exhaust on the enemy ADC to save his life if he can't react in time. Ideally this is a situation you should never get into, but if it does happen then try not to get mad at each other, and just calmly try to make the best of a bad situation.
Finally, what if you don't have voicechat? Talk as much as possible before laning starts about what your goals are, and then do as much communicating with pings as possible. You should chat the cooldowns on your major skills when the jungler comes bot and also once or twice a minute. Avoid criticizing unless it's something that they need to fix for later plays within the same game. And don't underestimate the importance of saying "Good job!" It can go a long way to increase trust between you and a random lane partner if not only do you do the correct things in engages but also acknowledge when they do the correct thing.