im currently DMing a campaign, that dispite only a few sessions, about 2 weeks has passed in total. 

My group will soon be travling to one of the major cities in the world and im concerend handling time. mainly if the members want to learn new abilites and train. So if for example the cleric in the group wants to learn an alchmists toolset, and for him to be a master at said toolset (in otherwords, gain profitenicy in the toolset, and not just know it) it might take 3-5 months (havent decided the time yet) how would you handle the time. 

Would you just say, "right so 3 months has passed your now a master in the toolset" or would you say he would have to train over multiple sessions in the city? And if i say the second option, how would i not make him board in a session, primarly when he is training for 8 hours a day. I dont just want to ignore him and tell him "your training, your not here" when the other characters are off adventuring. 

im aware of that section and its very useful, but im looking for other DM's experience. 

Basically you have two options:

1. Training while adventuring: 8h/day is excessive but you can rule that the player can devote 1-2 hours/day just before long rest (or as a part of it) to train. This will not disrupt the adventuring but it will take a LONG time for him to master the skill.

2. Time skip - if there is no immediate rush you can tell players that they should each declare what they would like to accomplish in a 3-5 month span with the provision that it can't be hardcore adventuring. They can do stuff together or individually and then meet after that time.

Both are fine but be sure to use 2. sparingly.

Oh, there is also the "substitute" route. Discuss if maybe the player has a substitute character in mind that could go on the adventure with the rest of the party while his main character stays in the city to learn.

I wouldn't have some members of the party adventuring while another is training (with the possible exception being if the party member knows they will not be able to play for a while.

If every member of the party wants to train in something (or do other downtime activities)  you can just waive the time away with a sentence and maybe a few rolls. Even if only some of the party have a specific objectiove the others migth be willing to work to earn their living costs while the resotof the party do other things.

Another way of dealing with it if the party have a long journey is to replace the instructor with books. Ther Cleric buys books on Alchemy and then whiel journeying he is in the cart studying while others are on lookout. If someone shouts of danger he can put the books down and help deal with the threat.

I like at substutute angle

Our group is in this situation now, and we do require training upon gaining a level.  People do have backup characters, and between a couple players leaving and new ones joining as well, the campaign unfolds like an anthology, which everyone enjoys.  So if time needs to pass for certain characters, it is very easy to either for one player to use a different PC or to focus on another part of the world.  We also do a downtime method, as everyone now has a week in town.  We are not having our next session until January, so this is resolved with a conversation thread for each PC.

For me, it's an issue of fairness. If one player gets a new proficiency, either everyone should be able to get one or gaining that proficiency should have a cost. And time itself has no cost until you give it one - it's trivial to wave your hand and say 3 months or 50 years have passed.

Personally, I'd let the party decide whether they wanted to take downtime as a group. In that period of time, they could focus on making money, gaining a proficiency, making connections,  researching spells, etc. Everyone gets to improve their character in the manner they choose (I'd have to put in some work here to determine the final bonus granted by each choice in a way that makes them roughly equal). They can do this whenever, but I'd make it clear to them that the world goes on. Unlike a game like Skyrim, your enemies will not patiently sit in their lairs while you skill up your alchemy. People die, towns burn, the world descends further into chaos and darkness. That is the cost of time in my campaigns.

I'll make an exception for retraining a proficiency since that is a lateral move. If the cleric wanted to switch out their herbalism proficiency for alchemy proficiency, I'd allow that to happen over a few sessions as they adventure, just describing their work as happening during rests and whatnot.

Check with the other players. If they’re into it, just hand wave the time. 
It could do something like, you have 4 months of downtime, what do you do? Maybe others also pick up a proficiency, maybe they just spend the time using one of their existing proficiencies to earn some money so they can buy that suit of plate mail. Then the time ends. 
You could do a session (Or part of a session, or via email between sessions) timed while this is happening that’s purely RP. Where, as they are doing whatever they’re doing, they meet some new people who become NPC friends or mentors or rivals. Or they start hearing rumors that lead them toward their next quest. 

My DM has a system for learning new skills and abilities while traveling, since she prefers the kind of games where the characters are constantly on the move, but also doesn't want to ignore the options to learn new things or craft items.

Here's the system she created, which isn't super balanced for those who aggressively min-max, but works for a group like us that focuses  a lot on roleplay. She determines whether a skill is simple, medium, or complex, each with a goal of 100, 200, or 300 respectively. As part of a long rest, a player can roll a d20, and add that number to a cumulative score, and when they reach the goal they gain the new skill they're researching. The Caveat is that they either need to have someone with proficiency in the skill they're studying to teach them, or they need to find a book on the subject to study from. This is most frequently done, in our games, with languages. Most languages are considered a Medium challenge to learn, while learning a new tool is usually complex. There are rarely simple challenges... most often that's done for crafting objects too complicated to simply craft in an hour or two.

When it comes to long periods of downtime, however, I do think it's best to just kind of skip over it, but only if the entire party gets the same length of downtime. When I'm working as DM, if the adventure allows an extended period of downtime, I basically just tell my players "You get downtime for X amount of time... what do you do with your time?". And from there I just use the downtime rules from Xanathar's to decide what happens. Not ever one of my players is particularly interested in downtime... while one player might carefully budget their time and resources to gain magical items or study a new skill, another player might just say, "I spend a week gambling", and I have them do a few rolls to determine how much they make in that period of time. I am, however, careful to let them know at the end of a session something like, "Oh, by the way, you're all going to have a month of downtime after this", so that they can look up what they want to accomplish in that time. Most players don't regularly think about downtime activities, so it helps to give them time to prepare, rather than springing it on them in the middle of a session.