In your experience, how frequent is best for levelling up? Obviously, it's exciting to level up and it allows for greater player agency if you have it more often, but go too fast and the players don't have time to become attached to each stage of their character.

Published adventures seem to aim to level you up after your first quest and then around every 2-3 quests after that, at least all the ones in my experience have done that. How do you feel about that pace? What pace do you think is best? What benefits have you seen from that pace?

I would say the first one after 1-2 hours then every 6-8 hours of play feels about right.  How much that actually covers in game will vary wildly between groups and adventures.

Published adventure pacing *feels* a touch accelerated to me, but is probably the most balanced between advancement to allow for combat with the narrative BBE and allowing players to become familiar with the new abilities that they've gained. I feel like players don't always grasp their abilities as quickly as level advancement in published adventures allows. Once the party hits 4th level, I generally back this off to every 2-3 sessions that contains a majority combat or 4-5 sessions otherwise. 

If you don't want to use XP, or have levels be at set milestones, you could have the party level based on the number of sessions played.  I was in a game of DCC where they did that:

Level up to L2 after 2 sessions with their L1 characters.

Level up to L3 after 3 sessions with their L2 characters.

Level up to L4 after 4 sessions with their L3 characters.

etc.

That method at 1 session per week, takes a party from L1-10 in roughly 1 year, or from L1 to 20 in a bit over 5 years.

A benefit of leveling taking longer at higher levels is that it gives the DM more time to adjust to the party's new abilities so they get appropriate challenges but can also face opponents that were once challenging that they can just blow through, highlighting how much stronger they've gotten.

I use this a good planning tool.

DnD 5e - Practical Guide to Campaign Planning | RPGBOT

Dividing the XP required to level by the XP for an average encounter gives a fairly consistent number of encounters to level.

Levels 1and 2 require 6 medium encounters to give encough XP to level up.
If you can get through 3 encounters in a session (should be doable at low levels) then it will be four sessions to get from level 1 to level 3.

Level 3 requires 12 medium encounters to go up a level.

Levels 4 through 9 require 15 medium encounters.

Level 10 to 11 requires 18 medium encounters. Not sure why the big jump here. There are a lot of features at level 11, it seems like it would have been better to spend more time from 11 to 12 than from 10 to 11.

Levels 11 through 20 require 10 medium encounters. The second half of levelling goes by quite quickly, possibly to account for the fact that tier 3 and tier 4 encounters take a while to run at the table.

Answering this question depends a lot on your past experience with D&D.

In the game I am currently running, we've been playing D&D since 1e and the 5e pace of leveling still feels fast even if the characters are only level 9 after playing a year and a half. I've been leveling them using milestones at 1/2 or less the rate suggested in the modules. If I used the actual milestones in the modules they would likely be level 15+ by now.

Adventurer's League allows an optional level up after every session. So, someone used to playing AL may think that leveling up every 2-3 sessions seems quite slow.

Folks who play video games (depending on the games played) may have different expectations about how fast they should level up.

The bottom line is get an idea of the player expectations, then consider the material you have and how fast you want the players to engage with it then figure out how quickly you want them to level.

I would do level 1->2 in one session or at most 2 since the survivability goes up a lot.

After that, probably level every 2-3 sessions for an average group - optionally every session if running AL style content - perhaps less frequently than every 2-3 sessions depending on player expectations. It may also depend on how much time is spent role playing vs adventuring and whether the role playing is a part of the adventure or just interacting with other players and shopkeepers etc.

I've been running a pre-written adventure for the past two years, and I've been using Experience Points that whole time, partly because I almost never see people playing with XP and wanted to see what it was like. Honestly, the players pretty consistently leveled up right around the same time that the adventures recommended leveling up for milestone leveling. Although there were a few interesting moments where the players would level up right before taking on a particularly challenging area, which was always a big morale boost for the players.

In my solo campaign I'm running I am aiming for "when it feels right". I keep a tally of combat results, and the XP gained, but I also base it on how much the player is enjoying their character as they are. There's no need to level up if htey are enjoying themselves as they are, and it can actually be more of a detriment sometimes - they might be really enjoying the plan they're making, and then you go "oh, level up" and they have to remake the plan or an ability can invalidate the plan entirely, reducing the enjoyment of the players in this instance.

For a general campaign, I look to level up the players every 4-ish sessions (session lengths sitting at 3-6 hours), depending on how much they accomplish.  I will sometimes go down to 3 sessions if they have been particularly productive; sometimes 5-6 if they are being particularly roleplay heavy and not advancing the plot much (but are having fun with it, and don’t need a DM poke to get moving).

I think it is important to note that I will never start a player at anything less than level 3.  Pre-subclass, classes just feel rather vanilla, and I would not do that to my players.  If I were going to start players in the dull levels, I likely would likely accelerate their levelling so they can hit three by after the third session. 

While I like this method for the earlier levels, if you're getting into the higher levels, or even if you're meeting with longer intervals (my group meets every 2 weeks), even at L9, you'd be looking at 9 sessions for a level, which would be nearly 4 months to get from 8 to 9. 

Overall, yes.. I like it. But it definitely would need some scaling tweaks, depending on the campaign. 

I mostly go off when things start to feel a little stale, and you are getting the sense that your players want to try new things with their characters.

Get good at reading your table, and you can pick up on that mood before they're really aware of it and spring a level up on them a little early occasionally, as a surprise treat. 

That said, I usually also try to pair level ups with something significant happening in the game. Reached a new area, cleared a dungeon, defeated a boss, finished a mission, that kind of thing. Not that you should level up every time they do one of those things, but every time they do do one of those things you should ask yourself "when did they last level up?"

I shoot for around 10 - 12 major challenges (combats, avoided combats, "skill challenges," important social encounters, complex exploration scenes) which also typically includes at least as many minor scenes as well. This usually equates to 4-5 sessions.

I've also developed the habit of skipping level 1 entirely and "fast-forwarding" through level 2 with kind of narrated, interactive montage of scenes.

But ultimately it depends on what your group wants. My group is eager to get to characters with the complexity of a subclass, and beyond that they like a story pace that allows a full campaign to wrap over the course of about 2 years. I've built my pace around them and if I were to DM for a different group I'd get some feedback and try to fit their expectations.

But I will say I don't like a super slow pace. I want my narratives to have a feeling of rising danger. It's hard to do that if your group has been at level 2 for 6 months.

Considering that the exp breakdowns for the first few levels is so low early advancement will be quick.

It was a real shock for me when I saw only 300 exp for level 2 for everybody when it used to be in the thousands.

I do:

Reach level 2 end of session 1.
Reach level 3 end of session 2.
Level up end of every 3rd session from levels 3-6
Level up end of every 4th session from levels 4-9
Level up end of every 5th session thereafter.