[RPG] How to the Starter Set wizard have five skills

There are a few classes you can multiclass into to gain skills, such as Bard, Ranger or Rogue.

In addition to that, in the player's handbook there is a feat called "Skilled" which allows you to gain proficiency in any combination of 3 skills or tools.

In Xanathar's Guide to Everything, we have the Prodigy feat which gives a skill proficiency and a few other things, but can only be taken by a Human, Half-Orc, or Half-Elf.

In Tasha's Cauldron of Everything, the feat Skill Expert gives proficiency in a skill and a few other benefits, with no racial prerequisite.

There is also a bard ability in the college of lore that allows you to learn 3 more skills at level 3.

Currently, the downtime rules allow you to gain proficiency in tools but not skills.

In case it isn't clear, you don't have to pick skills from your class skill list when you gain new skills through the "Skilled" feat. You can pick any skill you want.

This is because certain weapons break these general rules. Also, class features which are not obvious without the construction rules.

The attack for melee weapons defaults to Strength. But if the weapon is a finesse weapon (like the wizard's short sword), the attacker may choose to use Dexterity instead. So the Wizard, who has a much higher Dex than Str, uses Dex for those attacks.

With ranged weapons there is a similar exception. If you use a Thrown weapon, you may choose to use Strength instead of Dexterity. Thus the melee Fighter's javelin attack uses Strength instead of Dex.

The last question is a bit different. The Ranged Fighter has a class feature that provides +2 to hit for ranged weapon attacks. So they get Dex + proficiency + 2 for +7 total

Hope this helps, and please feel free to ask about anything else you are stuck on! If you haven't already it might help you to read the full basic rules documents located on WOTC's website.