The actual class tables in question, like the one on p. 113, say variations on the following:
Hit points at Higher Levels: 1d6 (or 4) + your Constitution modifier per wizard level after 1st
This suggests you add your CON either way, and is IMO much less ambiguous than the wording in Chapter 1.
Current hit points are unaffected by hit point maximum increases
A character's hit point maximum is only the upper limit on the creature's current hit points:
A creature's current hit points (usually just called hit points) can be any number from the creature's hit point maximum down to 0.
This is reinforced in the section on healing:
When a creature receives healing of any kind, hit points regained are added to its current hit points. A creature's hit points can't exceed its hit point maximum, so any hit points regained in excess of this number are lost. For example, a druid grants a ranger 8 hit points of healing. If the ranger has 14 current hit points and has a hit point maximum of 20, the ranger regains 6 hit points from the druid, not 8.
Things that increase your hit point maximum (or remove an effect that was reducing your hit point maximum) have no effect on your current hit points unless stated otherwise.
This is reinforced by the wording of the aid spell:
Each target's hit point maximum and current hit points increase by 5 for the duration.
Note that aid separately increases your max HP and your current HP. If increasing your max HP automatically increased your current HP as well, it would be redundant and potentially confusing to state both parts separately. However, because your max HP is just an upper limit, and increasing it doesn't increase current HP automatically, it's necessary to state in the spell description that both are increased.
In short: max HP increases don't change your current HP unless stated otherwise (as aid's description does).
(Note that hit point maximum decreases can decrease your current hit points, because your current HP can never be above your max HP.)
Best Answer
This depends upon which option the DM is using.
The 5E rules have 2 official variations:
Note that average for a die is (minimum + maximum)/2. For all even dice, this ends in a half (a _.5). The same effect can be done by dividing the maximum, then adding 1... 0.5 for the average, and 0.5 for the round up...
Standard
If the stock option is used: at level 1, take the type of die (eg d8) and you get the number of sides + your con bonus (so if mr d8 HD had con 10, he'd get 8). Each level therafter, you roll the HD and add it to the maximum.
Option (and Adventurer's League)
If the stock alternative is used... At first level, still the number of sides plus Con bonus. (Number of sides)+(Con mod). At later levels, 1+((number of sides)/2) + (Con mod) for each level.
Multiclassing
Only the first class taken gets "1st level HP"... if you multiclass into another, you still roll (or take the reduced amount.
Handy Chart
Note that only monsters use d4 and d20 HD. Con Mod multiplied by level is added to the total (effectively it modifies each die).
Common other variations not in the rules
Rolled at level 1:
just roll the die each level, add con bonus. Not common anymore.
Rolled at 1 but with minimum.
Roll the hit die at level 1, but if less than the variant's listed HP/die, raise it to that. So, fred is level 1, and rolls a 6 on his d10. He keeps it. Jojo also rolls, but gets a 2 on his d10. It raises to a 6. At higher levels, most DM's using this option do not allow raising.
Some Examples
Fred Fighter
Fred is a fighter. He's level 3. He's using the alternate, because he's in an Adventurer's League Game. He has a Con of 14, for +2.
Billy Barbarian
Billy's in a campaign using the standard rules. He's 6th level, and Con 16 (+3)
Charlie Changer
Charlie is 6th level... but went Wizard to 3rd, then fighter to third, then cleric for 7th.. he's also con 8 (-1)... Campaign is using the alternate only.
Note: Con penalty applies every level, too. And at level 4, the character is level 4, so only gets the per level "fixed roll values"