First, you gain hit dice (and a corresponding increase in maximum hit points).
This is described for each class briefly under Class Features, and explained on page 15 of the PHB (or here in the free Basic Rules):
Each time you gain a level, you gain 1 additional Hit Die. Roll that Hit Die, add your Constitution modifier to the roll, and add the total to your hit point maximum. Alternatively, you can use the fixed value shown in your class entry, which is the average result of the die roll (rounded up).
Almost everything else is shown on the table in each class description.
For example, page 71 for the fighter or page 113 for the wizard. You can see the charts for the basic classes online for free as part of the Basic Rules — for example, the Cleric Table.
As you go up in level, if a value goes up on the table, use the new value. (The plusses are not cumulative.) All classes have a proficiency bonus which works this way. When you increase this value, of course, also increase all the places on the sheet like trained skill checks which depend on it. Other examples of values which increase on the chart are the number of cantrips known, the monk's Unarmored Movement, the sorcerer's Spells Known, or the barbarian's Rage Damage.
And, if you get new spells slots, they'll be shown there too.
Then, look in the "Features" column. Anything there will be described in the corresponding description later in the section on that class. In general, when these class features improve at higher levels, there will be a note on the table. For example, Druid has "Wild Shape" at level 2 and "Wild Shape Improvement" at level 4.
Some features have per-level scaling not mentioned in the chart.
For example, the number of distinct spells a wizard can prepare per day (which is separate from spell slots) is equal to Intelligence modifier plus wizard level — but note that this isn't really something you need to know when leveling up per se. It's something to know when preparing spells. A Moon Circle druid's limit on creature CR is similar: at 6th level, this becomes druid level divided by 3. You don't need to know this when you level up, just when you shift into beast form — but of course it's good to have this in mind so you can be prepared at the table.
Other increases do basically happen at level-up time. For example,
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free.
So, really there's no help for it but to be aware of which of your class-features are level-based. Most features which have something level-dependent not on the chart also have an additional complication. For example, wizards can add spells as above, but also in other ways, so just having the chart show number of spells known wouldn't work for wizards the way it does for sorcerers and bards.
The good news is that there's not really a lot of these, and the ones that exist tend to be core to the way your character works at the table (like, wizards and spellbooks or moon druids and shapeshifting) so they're relatively easy to remember. And they are all described in the appropriate class features section, not scattered randomly throughout the rules.
A tip: mark these in some way on your character sheet (with a ⛤, for example). That way, you don't have to scan through all of the features every time.
Some racial traits get better.
This is also just something you'll need to look out for. Usually, it's when you have the ability to cast spells — you may get more at higher level. In other cases, damage increases.
In the Player's Handbook, this only applies to the Drow Elf subrace (spells), Dragonborn (breath attack), and Tieflings (spells). Oh, and Hill Dwarf gets an additional hit point per level.
As with class features, marking features which scale with some symbol you'll remember on your character sheet might help.
Damage from most attack cantrips goes up at certain points.
These are "level 0" spells — they don't use spell slots or preparation and if known can be cast at will. Most damaging cantrips increase in power at 5th, 11th, and 17th character level. This is noted in the spell description, not on the class chart, because it scales with character level not class level (which may differ if you multiclass — see below). If you have cantrips written on your character sheet as attacks, this is good to be aware of when you level up.
It's also important to realize that other spells don't do this. Some spells note that they can be cast using a higher-level spell slot and give what the effect of casting in that way would be. For example, Magic Missile says:
At Higher Levels. When you cast this spell using a spell slot of 2nd level or higher, the spell creates one more dart for each slot level above 1st.
I can see how it'd be easy to interpret this as meaning "2nd character level", but D&D uses "level" to refer to different things (in 5E, character level, class level, and spell level are all distinct). Here it refers to spell level — Magic Missile is a first-level spell, but if you cast it using a higher-level slot, you can create more darts. From the class chart, you can see that wizards gain 2nd-level spell slots at 3rd class level. A 3rd-level wizard has two 2nd-level slots and four 1st-level slots, and so could cast magic missile six times, with two of those times gaining an extra dart.
In general, except for cantrips, spell damage (or number of targets, or etc.) does not increase as you level up. Instead, you gain higher-level slots, which you can use to cast some spells with greater effect.
Pretty much nothing else changes.
D&D 5E uses a concept of "bounded accuracy", which means that few things scale just for the sake of going up as you level. You mention "level 2 attributes and passives". This isn't a thing. At 2nd level, your attributes stay as you chose originally (unless of course the DM lets you rebuild; in Adventurer's League you can rearrange things at will up through level 4).
Ability scores only increase as part of the "Ability Score Improvement" class feature (or through Feats which grant increases). When you do get to the point where you increase an ability score to the point where the modifier changes, saves and skill modifiers (including passive checks) should be adjusted to match — but this is technically part of the Ability Score Improvement class feature, not of leveling alone.
If you are multiclassing — mixing and matching between different classes as you level — special rules apply. (See Multiclassing in the Basic Rules.)
If you are using the Starter Set...
In this case, the included rules don't include a generalized guide to classes or leveling up. Instead, on the second page of each character sheet, there is a section called Gaining Levels which tells you exactly what increases for each sample character at each new level. Do exactly what it says, and no more. These character sheets cover everything you need to know up through 5th level, but not beyond. For example, since the proficiency bonus increases at 5th and 9th level in the standard rules, the example sheets note this as increasing at level 5.
This is all fine, but I highly recommend at least adding the Basic Rules, downloadable from Wizards of the Coast for free, or available online (ad-supported free) as part of D&D Beyond. This will help you better understand what's going on, and give you more options. (And, for that matter, the PHB costs less than a gaming group all going out to a movie one night, and lasts a lot longer.)