Important notice on mods: Avoid DT and HR until you're good(good like in 3000 pp). While HR seems like a possible and challenging mod in your early stages of playing, both mods will soon lead into bad habits or become plain impossible to play properly(AR7+HR/AR8+DT=Death for inexperienced players that have yet to master some basics).
It is ok to toy around with every mod for a bit but the ability to play nomod is the fundament to being successful with every mod. Therefore you should try to play more nomod maps than maps with mods. SD/PF are especially counterproductive to improvement as they are about not playing the game. If you want to use them anyway, this guide is not for you.
EZ/HD/FL are the least detrimental mods to learn early and I think it even makes sense to take a look into EZ/HD relatively early but they require good nomod-fundamentals too - try them on maps you already fullcombo'd nomod. HT on maps out of your range is okay but usually unnecessary as ranking criteria ensures the availability of easier maps for the same song.
tl;dr play nomod

Step One:
Go into your options. Activate the FPS-Counter. Set your Framerate to Unlimited. Play a random map and keep checking the FPS. If it's constantly below 2ms(or above 500fps on the Fallback-version) keep it on unlimited, otherwise change it to 240 FPS. This recommendation comes from the developer himself and has something to do with osu!s weird code. Don't question it. You can turn the FPS-Counter off afterwards if it annoys you.
Do other stuff that improves your playing environment like getting a clean skin and 26476 other possible things to achieve this. I just mentioned the FPS-thing cause I personally didn't know it for a while and it didn't appear in the guides I read.

Step Two:
Download Beatmaps. Like 1k at least okay? It's fine to search for songs you like but you won't ever get enough maps to properly practice on. Abuse this, this and this to get a bunch without running in your download-restriction. As mapping changed over time I'd suggest you to take a couple of packs out of each timerange to get some insight on different mapstyles.

Step Three:
You know what Collections are?
If not, go into your map collection and rightclick any map. Choose "Manage Collections".
At the top there'll be a box that says "Select a collection or create a new one here".
I will tell you some collections you should make for my approach. The idea is to organize every single one of your maps in accuracy brackets.
I suggest the following collections:

<88%
88-92%
92-95%
95-97%
97-98%
98-99%
99%+

This has multiple advantages:
First of all they will always be on top of your collectionlist, making it easy to sort maps into the correct collection right after playing them.
Second is that the amount of badness you're showing on a map gets evaluated more rough the worse you performed. Getting 88,31% or 90,67% Accuracy on a map isn't a real difference(unless it's a streammap), it's both pretty bad and can fluctuate easily with 1-2% while the difference between 98,3% and 99,3% is a lot easier to grasp and not as inconsistent in most cases.

Step Four: Now filter your maps "By Rank Achieved"(this will keep you from scrolling through maps you don't have a score on) and sort all of your maps into the collections. If you're the newb at the game who wants this kind of instruction it will only take a few minutes. If not, oh well. Take your time, spread it over some days or be creative.
Also don't cheat yourself. If you got 91,99% on a map it doesn't belong into the 92-95% bracket. You have to make the cut at some point, be clear with it. Your best score counts, not your best acc. The reasoning for this is that a worse score implies a combobreak implying that you lost control at a point where you were stable before even though you might have had worse accuracy.

Step Five:
Create other collections that are sensible for you. People tend to have separate collections for stream-maps, maps they like, music they like(yes there's a difference between these two), warmup-maps, things they want to fc etcpp.

Step Six:
Play more. There are two kind of plays you want to make everytime you're playing osu! (to improve):

Step Six point One:
The first is sightreading maps you didn't play before.
Use a rather broad difficulty range here. As the maximum difficulty I would recommend a stardifficulty you're confident that you can get a B on though.

To find such maps you can set filters by typing "star>a.bc star<x.yz" and then happily press F2 or filter by difficulty and work your way through(If you downloaded a lot of maps you can skip songs where the preview sounds disgusting to you).

Especially if it's a map that is hard for you(considering star rating) I strongly recommend using the NoFail-Mod. You don't know if the map contains patterns you're horrible at and when talking about sightreads it's about playing the map through and seeing what you get. Even if you happen to have no problem passing the map when actually playing it, it doesn't hurt either and NF makes it very easy for you to beat that score later in case you got a highcombo-lowacc-play or something similar.

Sightreads shall improve your ability to read all the patterns osu! has to offer and even while you might get some fullcombos you can easily achieve them later on again(also NF only gives a 10% pp-penalty which is pretty much nothing at a low pp-count).

Remember to put every map you played into the according accuracy bracket right after finishing it. It's a good habit.

Step Six point Two:
Now the work with accuracy brackets is coming into play.
Filter your maps by Collections and Difficulty.

Your aim should be to move some maps into a higher accuracy bracket everytime you're playing. Focusing on the 88-97 range ensures that the maps you're playing are challenging you in some way while playing maps in the 97+ brackets are suited to practice pinpoint accuracy and consistency.
Choose a bracket first.
Select the easiest map from the respective bracket, play it no-mod and try hard for your accuracy. However a rule of thumb:
Don't play maps from the accuracy-brackets more than once in 3 days - assuming you're playing everyday, make sure you played a reasonable amount of maps inbetween otherwise - if you want to play a specific map more often put it in your like-collection or something like that. Trying to fix a score while fixing accuracy leads to a lot of retrying without much benefit.

If you find that you failed to move 2 or 3 consecutive maps in a bracket you should go to another bracket or go for sightreads as it is very unlikely that you will have success on improving your acc on even harder maps of that bracket then.

It is up to you which brackets you attend the most but you should try to look into each one once in a while(99+ excluded). Personally 88-95% is most satisfying for me but I guess that's part of personal preference.

The great thing about this is: Due to the fact that you won't be playing the same maps all the time you will improve on the other maps you're playing between two plays of the same maps meaning that it is very likely that you will be able to easily move the map into a higher accuracybracket with one try(that's motivating you know, psychology).
The other thing is that you will be playing the maps with a low star rating that are hard for you more often than others. Therefore you will inevitably tackle your problems and become a better player as a result.

Depending on the amount of maps you're shipping around in your collection you will find that a lot of 1-2 star maps pile up in the 95-97 and the 97-98 bracket especially when focusing on the 88-92/92-95 brackets. In the <88%-bracket you will usually find all the maps that are too hard for you and you miserably failed on and the close ones with 86-88% will quickly move into the 88-92%-bracket.
It's okay to leave the maps in the 95-98% range dying and scroll down to harder maps if they are noticeably easier than your worst topplay unless you played them recently. If they become too many and it is a bother for you to scroll down you can make collections like "97-98% garbage" and put them in there so they're out of your way. Don't do it lightly though. There is usually a reason why the map didn't move up to 98%+. If you're unsure play the map.

Over a long time there will also appear some easy maps in the low brackets that seem to be outright shit and extremely hard to acc(mostly very old maps) which is why they aren't moving up. Let them rest for a longer time than you would with other maps, even they will get better with time. Some of them are shit indeed but others will only look like shit and suddenly look reasonable once you can play them properly. And even the bad ones are usually worthwhile mastering. If you can get good accuracy on something that doesn't make sense it means that you mastered the concept of that mapping technique as you're able to apply the sense yourself upon reading.

Step Seven:
Boost your pp by selecting specific maps that you played on the sightread-tour that you found enjoyable or easy in relation to the star rating and try to FC them with good accuracy. Having a wide variety of maps you played once gives your pp-efforts a straightforward direction and results in less time spent per pp-play.

If you're not into pp you can skip this step and play your favorite maps with Hidden, Flashlight or Easy to fully master them or whatever.