I'll do my best to answer your questions and fix some of what Shroudborn said (not tryin to be mean just some corrections, other people feel free to correct me if I'm wrong).
First of all, the "sliders" are more popularly called long notes or abbreviated to LNs, though most people would understand what you are talking about if you used the "slider" terminology. For LNs, you have to press the beginning of the note and release at the end of the note, nothing fancy about it. Most players starting out have an issue with these types of notes and continue to have issues even when they begin to "git gud" at the game and develop a hatred for them in general (plz dont hate my babies
) As for the best judgement you can get (i.e. a rainbow 300) we call those MAX's and the other 300 we just call 300 (why did they do it this way?). You don't have to hit a note exactly on a perfect frame to get a MAX, you have some leniency. I will use OD 8 as measure of this:
MAX timing window: + - 16.5ms
300 timing window: + - 40.5ms
200 timing window: + - 73.5ms
100 timing window: + - 103.5ms
50 timing window: + - 127.5ms
And a miss is a miss, or a note hit way too early.
As you increase or decrease OD, the timing windows will get tighter or more lenient respectively. The timing windows aren't necessary to know to hit a note correctly, this comes intuitively as you play the game more, so you'll get the hang of it eventually, or you might not at all, like me who sucks at accuracy.
No other mods give bonus pp other than DT currently (please God keep it that way, and remove DT pp). As to how DT amplifies your pp, I'm not sure, but lets just say it's a lot. I think it just takes the star rating of the song you DT'd and calculates it as if it were a normal map (I have no clue). DT does not affect score in any way, nor do the other mods.
First things first, make sure you have BPM scaling/ scroll speed remembering off in the settings. This is located in the gameplay tab in the options, or the 3rd circle looking thing. This will allow you to find a comfortable scroll speed that you can get accustomed to and will greatly help your accuracy. As for tapping speed, my fingers are probably just as fast as your's. The main difference in skill here is reading ability, or the ability to see the patterns that the notes flying down (or up I don't judge) at you are. This comes with getting muscle memory down as you play through patterns on a multitude of different songs. DO NOT "practice the living daylights" out of maps you like or want to get better at. This will lead to mindblocks and you will probably be unable to play that map later on even when you are skilled enough to do so. Instead, focus on improving accuracy on songs and trying new things you have yet to attempt. Everyone improves differently, so find the method that works best for you, but notable methods include: S'ing songs you have A's on, A'ing songs that you have a hard time on, etc.
Goal songs are a great idea, and have been really beneficial in my own improvement at this game. The way I do this is I break up what songs I want to pass in the near future, and songs I want to pass a month or two from now. I've gone so far enough to set ultimate goals for myself to see if I can reach them one day. Point is, this is a great and relatively fun way to improve.
Play with people of all different skill levels. You will probably do most of your playing with people around your skill level, but that doesn't mean you should constrict yourself to this. Playing with higher level players might yield you some great tips at how they have experienced and made progress in the game that you can apply to yourself. Playing with lower level players can lead to some fun times as well, I wouldn't discourage anyone from doing it, as in the end, multiplayer is for the people not the competition in my opinion. You should also make some rivals of some sort like Shroudborn said, the competition between friends is very entertaining and pushes you to do your best.
If you can find someone to coach you, you'll be luckier than most. I've never had a coach, but I can tell you that a great substitute is playing with higher level players like I stated before. Many don't mind answering any questions you have about the game and want to see as many people improve as possible.
Jakads is pretty good ye