Tabaxi have a climb speed of 20 feet, and a normal movement of 30 feet.
So let's say I move 10 feet to a wall. Can I then climb 20 feet for a total of 30?
Another way to look at this is your climb speed is 2/3rd that of your walking speed. So climbing 20 feet would cost me all 30 feet of movement, which I think it does. But what if you walk 20 feet to a wall that's 10 feet high? Can I climb it?
Now to add to the confusion, what if you're using the Tabaxi's Feline Agility? Doubling my movement. Now I have 60 feet of movement. Does my climb speed become 40?
Now let's add more confusion and say I'm a Barbarian with fast movement, which adds 10 to my movement speed. How do you calculate that all out?
Maybe the simple solution is to pretend the wall is x1.5 higher than it normally is: 20 x 1.5 = 30
Tabaxi have a climb speed of 20 feet, and a normal movement of 30 feet.
So let's say I move 10 feet to a wall. Can I then climb 20 feet for a total of 30?
Another way to look at this is your climb speed is 2/3rd that of your walking speed. So climbing 20 feet would cost me all 30 feet of movement, which I think it does. But what if you walk 20 feet to a wall that's 10 feet high? Can I climb it?
Now to add to the confusion, what if you're using the Tabaxi's Feline Agility? Doubling my movement. Now I have 60 feet of movement. Does my climb speed become 40?
Now let's add more confusion and say I'm a Barbarian with fast movement, which adds 10 to my movement speed. How do you calculate that all out?
Maybe the simple solution is to pretend the wall is x1.5 higher than it normally is: 20 x 1.5 = 30
This means order matters, RAW. Using your Tabaxi example, you could climb 20 and then walk 10, but you could not do what you asked about, walk 10 and climb 20, because after walking 10, you only have 10 feet of climbing left. However, it is very common to houserule this (in particular, because tracking the order you move in is a hassle) so that instead, your Tabaxi would have 30 total movement, up to 20 of which could be spent climbing, in any order. But this is a houserule - I just gave you the actual RAW and answered your question using it.
EDIT: I just noticed your second question. As a general rule, anything that affects "your speed" is interpreted to mean "all of your speeds", although I am not aware of this being codified in any rule anywhere. This means Feline Agility is interpreted, by consensus, to double both your walking speed and your climbing speed. Same answer for additive movement (you asked about Barbarian fast movement, which adds +10) - if something doesn't specify a speed and just says "speed", it's assumed to mean all of your speeds.
The PHB covers how to handle using multiple different movement speeds in a turn. Basically any movement you take during the turn is subtracted from all your movements at the same time, including modifiers for things like difficult terrain. So in your example after moving 10ft up to the wall your Tabaxi would have 20ft of walking speed and 10ft of climbing speed remaining.
As for Feline Agility, or Fast Movement they both simply say that your speed increases and do not indicate a movement type. So they would affect all your movement types. I'm not sure of any examples of a movement speed increase that specifies type, so I can't give a good counter example.
Note this is the official rule, but now order matters.
That is, say you have a Climb of 20 and a move of 30.
You can climb 20 ft up and then walk another 10 ft. But if you walk 10 ft, then you can only climb 10 ft up.
Does not make sense to me, but that is the official rule.
I'm betting it doesn't make sense to them, either. But I don't think they can make it make sense, and make the rule simple enough for everyone to understand.
With a 20/30 ration Climb/Walk you can essentially make every other square (on the climb) 10 feet instead of 5 feet. And it "slows" your movement down the correct amount. But then there are many different ratios.
Pro rating the movement by a percent seems simple to me.
I.E. If you use half your climb move, you have half of any other speed left.
That is, if you climb 10 of your 20 climb speed, you have used up half your movement and can only walk 15 ft, regardless of other factors. Round up at .5.
5 climb leaves 3/4 of your 30 or 25 ft walking (same as existing rule) 15 climb means 10 of walking left (rather than 15 using existing rule).
But not everyone playing has graduated high school math.
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Tabaxi have a climb speed of 20 feet, and a normal movement of 30 feet.
So let's say I move 10 feet to a wall. Can I then climb 20 feet for a total of 30?
Another way to look at this is your climb speed is 2/3rd that of your walking speed. So climbing 20 feet would cost me all 30 feet of movement, which I think it does. But what if you walk 20 feet to a wall that's 10 feet high? Can I climb it?
Now to add to the confusion, what if you're using the Tabaxi's Feline Agility? Doubling my movement. Now I have 60 feet of movement. Does my climb speed become 40?
Now let's add more confusion and say I'm a Barbarian with fast movement, which adds 10 to my movement speed. How do you calculate that all out?
Maybe the simple solution is to pretend the wall is x1.5 higher than it normally is: 20 x 1.5 = 30
If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move. Whenever you switch, subtract the distance you've already moved from the new speed. The result determines how much farther you can move. If the result is 0 or less, you can't use the new speed during the current move. For example, if you have a speed of 30 and a flying speed of 60 because a wizard cast the fly spell on you, you could fly 20 feet, then walk 10 feet, and then leap into the air to fly 30 feet more.
This means order matters, RAW. Using your Tabaxi example, you could climb 20 and then walk 10, but you could not do what you asked about, walk 10 and climb 20, because after walking 10, you only have 10 feet of climbing left. However, it is very common to houserule this (in particular, because tracking the order you move in is a hassle) so that instead, your Tabaxi would have 30 total movement, up to 20 of which could be spent climbing, in any order. But this is a houserule - I just gave you the actual RAW and answered your question using it.
EDIT: I just noticed your second question. As a general rule, anything that affects "your speed" is interpreted to mean "all of your speeds", although I am not aware of this being codified in any rule anywhere. This means Feline Agility is interpreted, by consensus, to double both your walking speed and your climbing speed. Same answer for additive movement (you asked about Barbarian fast movement, which adds +10) - if something doesn't specify a speed and just says "speed", it's assumed to mean all of your speeds.
The PHB covers how to handle using multiple different movement speeds in a turn. Basically any movement you take during the turn is subtracted from all your movements at the same time, including modifiers for things like difficult terrain. So in your example after moving 10ft up to the wall your Tabaxi would have 20ft of walking speed and 10ft of climbing speed remaining.
As for Feline Agility, or Fast Movement they both simply say that your speed increases and do not indicate a movement type. So they would affect all your movement types. I'm not sure of any examples of a movement speed increase that specifies type, so I can't give a good counter example.
Edit: I found one - Boots of Speed
Thanks, guys. That was a big help. I'm doing a one-shot and considering Tabaxi Beast Barbarian.
Note this is the official rule, but now order matters.
That is, say you have a Climb of 20 and a move of 30.
You can climb 20 ft up and then walk another 10 ft. But if you walk 10 ft, then you can only climb 10 ft up.
Does not make sense to me, but that is the official rule.
I'm betting it doesn't make sense to them, either. But I don't think they can make it make sense, and make the rule simple enough for everyone to understand.
With a 20/30 ration Climb/Walk you can essentially make every other square (on the climb) 10 feet instead of 5 feet. And it "slows" your movement down the correct amount. But then there are many different ratios.
Pro rating the movement by a percent seems simple to me.
I.E. If you use half your climb move, you have half of any other speed left.
That is, if you climb 10 of your 20 climb speed, you have used up half your movement and can only walk 15 ft, regardless of other factors. Round up at .5.
5 climb leaves 3/4 of your 30 or 25 ft walking (same as existing rule) 15 climb means 10 of walking left (rather than 15 using existing rule).
But not everyone playing has graduated high school math.