I tryied to search on Google and here but couldn't find any answer. This question/thread is regarding CAS sims.

I don't really like the way the game mechanics use for it where adults and elders has to start from the scratch when you create them in CAS when learning skills and badges, so I usually I randomize their skills using College Adjuster, Simblender and InSim. Same thing with premades because even if in Vanilla, it isn't that hard to earn the normal skills without even trying (such as painting, cooking etc), so I always frustrated how the game think, when the adults has less skills than their children. (Sure, some sims may be stupid and lazy and do not choose to learn anything and focusing on other game aspects).

As I tends to skip sim days when playing, I usually cheat their skills with these hacks rougly based on their age and hobbies. But some of these hacks gives you way too many skills/badges so I usually do it manually. With Skills I typically asign about ½ to 1 skill per day of a sims life (A sim who is 28 sim-days which in my game is an adult usually have 14 to 30 skills points) and give them less and less as they grow older to simulate the slower learning it takes when your sims progress to maximize the skill.

But recently gone a bit tired of this method but I try play the game without cheating from time to time, but still assign some skills, badges, hobby enthusiasm for my CAS. Particularly with Badges, I still trying figure out how to assign them to my CAS sims without giving them too many/little (looking for a middle ground), so I thought about the time learning them, but couldn't find anything when searched online.

Taking completely vanilla, which I have played for a long time - and times roughly estimated, because my sims still have to study, they don't get skills for free.

Badges don't take quite as long as skills. (Edit - actually, it goes a lot faster).

Skills - especially mastering all of them (which is in my opinion the hardest LTW to fulfill in a vanilla game) - can take a lifetime for a new adult CAS Sim and he still may not make it.
The first 3 - 5 skill points are quick - from around 30 sim minutes to a number of hours - but then it gets harder. I think from 6 you are talking days. Every skill takes longer to master than the previous one and for the 9th and 10th one it is easy 4 - 6 days (of basically doing nothing else but eating and sleeping).

Obviously, if the sim is married, work, helps to raise kids (and knowledge sims are good here, they love to teach toddler skills, help with homework, etc). - that gives them less time to study. I do feel that social sims take even longer, but that may be because they just have to phone somebody or something, interrupting their studying, to not go into aspiration failure.

Knowledge Sims with the applicable aspiration point can have an Eureka moment at times and master a skill slightly quicker.

Some aspiration rewards help, of course - but they can only use those while in gold or platinum status.

Of course, if you start to build their skills as toddlers, and they have some skills when leaving school and then go to Uni, it is easier - but the last skills will still take several days to master.

It might just be me, but I feel like maxing all the skills is quite easy. Even for a CAS adult Sim, if they don't have a job, I can fulfill that want before turning to elder. If they do, it may take them a little longer, but nothing too crazy. For a born in-game Sim, if I really wanted it, they could have them maximized before adulthood (one time, a Sim of mine maximized Creativity as a toddler!). I usually don't, so, if they're going to max them it will be during their adult and elder stages.

I actually think badges are harder to learn. Well, some at least. I hate, hate, HATE the cashiering skill (note to self: get the Practice mod!). It takes sooo long to even get bronze, and bronze is not nearly enough to keep customer's from a somewhat busy business happy. For me, Robotics is also a tedious badge to gold. On the easy side, though, Sales and Fishing are really easy. Gardening too if you have some garden plots. Trees alone won't do the trick.

I play vanilla lifespans and skilling times.

For new CAS adults I first use the simblender and do randomise skills, then I use the sim manipulator to take down each skill past 5 back to 5. Any skills lower than 5 get left as is.

In a vanilla game this depends. Learning of skills is influenced by other things, it isn't just a set time. Some sims learn skills faster than others. If your sim is very lazy they will have a harder time learning body skill, they will literally tire out sooner and need to do something else while your active sim is still going. Badges in particular are linked to skills so mechanical skill will help with robotic badge.

If badges build faster than skills, they certainly don't seem to because there are only 3 milestones for each badge while there are 10 for each skill, so the Gold level badge always seems to take forever. It's a want slot clogged up with a locked want for that whole time as well, assuming I want my sim to actually get aspiration credit for completing the badge. I swear it's easier to get 30 best friends than it is to get a Gold badge in anything...since best friends are something a sim can get when I'm playing as other sims.

Some sims are of course faster at learning certain skills. A sim who is Neat, Outgoing, and Active will build Cleaning, Charisma, and Body faster than average. Serious sims build Logic faster while Playful sims build Creativity faster. There is, as far as I know, no sim who will naturally learn Cooking or Mechanical faster. Outgoing sims build creativity faster when performing on an instrument while Shy sims will build it faster by practicing on that same instrument. All sims build skills and badges faster in Fall.

Each skill point gets progressively harder. The first level of any skill will take barely any time, even Cleaning for a 100% Sloppy sim. The 10th point will take quite a lot of work, even if the sim has a boost from their personality for that skill (and even if it's Fall, etc).

Edit: Suppose I should add that I play with an aging mod that redistributes the number of days in each stage but that leaves the total number of days unaltered. I do not play with any mod that makes skilling either faster or slower, but I do use a mod that allows sims to build skills while on community lots. In my experience it is not overly difficult for a sim to max all skills, and most of my Knowledge sims will probably manage to max all skills before actually reaching the top of a career and certainly before reaching Elderhood.

Of course there are mods that can be used; I use some of them myself these days, in particular the community lot skilling one - but I don't think that is what the OP asked.

In the vanilla game, i turned aging off at times just to allow a Sim to gain that 10th skill point before his/her elder birthday. I remember one Sim standing in front of the mirror for three days in order to get that final charisma skill. I wanted him to get to the top of his career before the elder life stage for several reasons.

Badges and skills are a combination of many things (according to what I've seen, and the Prima Guide)
+ Personality - (neat better at cleaning, lazy better probably at writing)
+ Current skills ability - A sim with maxed out mechanics will get gold badges making robots faster
+ interests - Sim mostly likes nature will have lower interest in mechanics

(I'm not sure, but Prima says the enthusiasm they have for certain things also works as a booster. (probably in gaining skills and they way they love to concentrate on their favoured hobby object)

Do interest matter - yes in my game. Most in Riverblossom like nature, but there are a few who love talking about food and eating it. Consequently these sims are fat, because they also have no interest in sports or fitness. All my artistic sims are shy and messy. Guess so they don't mind the paint and are happy being alone when they work. Snooty Sims (http://www.snootysims.com/thesims2index.php?id=hobbies) and Prima guide (and others) have put information up about this.My playful sports loving sim won't sit still on the chess table to learn logic, and if I force him, he wines and cry's. So why make things harder for each other. He gets to play darts instead.

An unskilled sim, with no skills in said badge, whose personality hates it will take forever. Stick your outgoing girl or guy with tons of charisma on the till checkout and you'll see big improvements. Have you ever done well with either a mean sim or shy sim on the floor doing sales? Try it out and test it. Make ideals personalities go for skillig for badges, and make another sim who is unintrested do the same thing and see who gets to gold the quickest. And remember, there's always a sim that will just be different, because they is just difficult.