For your specific example, you add a total of +5 to your dice roll when you do a Strength (Athletics) check.
For skills and saves: do the math.
Rules don't ask for calculation untill dice are rolled.
From the ability checks rules:
To make an ability check, roll a d20 and add the relevant ability modifier.
[...] proficiency in a skill means an individual can add his or her proficiency bonus to ability checks that involve that skill. [...]
For example, if a character attempts to climb up a dangerous cliff, the DM might ask for a Strength (Athletics) check. If the character is proficient in Athletics, the character's proficiency bonus is added to the Strength check. If the character lacks that proficiency, he or she just makes a Strength check.
From the saving throws rules:
To make a saving throw, roll a d20 and add the appropriate ability modifier. For example, you use your Dexterity modifier for a Dexterity saving throw. [...] As with skill proficiencies, proficiency in a saving throw lets a character add his or her proficiency bonus to saving throws made using a particular ability score.
Whatever changes your ability modifier or your proficiency bonus before that roll, changes the numbers you add.
Wild shape affects both ability modifiers and proficiency bonuses.
From the description of the Wild Shape class feature:
Your game statistics are replaced by the statistics of the beast, but you retain your Intelligence, Wisdom, and Charisma scores. You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature’s bonus instead of yours.
You now use the creature's strength, dexterity and constitution scores (and thus modifiers), but keep your own intelligence, wisdom and charisma scores (and thus modifiers).
Regarding proficiencies, designers have clarified you gain and retain proficiency bonuses (and keep the best one):
Mike Mearls on Twitter:
You lose weapon proficiencies but keep skills. Use the higher of your or the creature's proficiency bonus
Jeremy Crawford on Sage Advice segment of Dragon Talk at 13:10, about a character who has proficiency:
You get to use your proficiency bonus. But you do use the creature's dexterity modifier. This is where it gets tricky. Use your proficiency bonus for anything where you're both proficient, but only if yours is higher, but you use the physical stats of the beast.
All in all, ability checks and saving throws are resolved the same:
D20 + ability modifier + best applicable proficiency bonus
considering your character's proficiency bonus is applicable whenever the character has proficiency, and the creature's proficiency bonus is applicable whenever the creature has proficiency.
How do I get the creature's proficiency bonus? It's nowhere in the stat block!
You use the creature's stat block skill/save value, and subtract its ability modifier.
You have to do this separately for each skill, as it may vary: some creature's skills benefit from a doubled proficiency bonus, when compared to the proficiency bonus per challenge rating table.
You may end up better than both your character and the creature!
- If the creature has better physical abilities, and you have a better applicable proficiency (e.g you have athletics proficiency, and turn into a polar bear).
- If you have better mental abilities, and the creature has a better applicable proficiency (e.g druid lacking perception proficiency with good wisdom, turning into a Giant Octopus).
For attacks: use the stat block
From the combat rules :
[...] To make an attack roll, roll a d20 and add the appropriate modifiers. [...]
When a character makes an attack roll, the two most common modifiers to the roll are an ability modifier and the character's proficiency bonus. When a monster makes an attack roll, it uses whatever modifier is provided in its stat block.
You add your proficiency bonus to your attack roll when you attack using a weapon with which you have proficiency.
When asked about attack rolls, designers point to the same rule. Jeremy Crawford on Twitter:
The intent is that the druid uses the bonus in the beast's stat block for any proficiency the druid lacks.
As you are using the creature's physical stats, and the creature's proficiency, the above formula matches most of the time - but flying snake's bite lacks the "finesse" property to keep it consistent.
I'd avoid recalculating attack scores entirely, even though it may sound tempting to druid/monks using unarmed combat in Wild Shape, or when shapeshifting in weapon-wielding forms. Let's keep these for a separate question.