[RPG] What skill or skills should apply to an appraisal check in 5e

First of all, D&D 5 treats language as a binary - you know it with absolute fluency or you don't know it at all. You could stop right there and it would be fair.

If you want to go further and allow someone to recognise a written or spoken language they don't know you need to decide:

1. The applicable stat - Intelligence is a clear stand out
2. The applicable proficiency, if any - we'll come back to that
3. The DC - clearly this should be based on how prevalent the language is in your game world; I'm not giving any advice on that but guidelines are in the book, they go from easy to damn near impossible

Which proficiency is appropriate?

Whatever one the player can make a reasonable case for

Depending on the language, I can see a case for:

• Arcana
• History
• Investigation
• Religion
• Nature
• Insight
• Perception

However, I'm not the lawyer for the prosecution; it's not up to me to convince you. Let the player state their case: if they convince you it applies, if they don't it doesn't.

Rules for Identifying items are found in the DMG, though they can also be found in the basic rules, which I will be referencing here. The basic rules are free to use and can be a good substitute until you can buy your own DMG:

Dungeon Master's Basic rules, pg. 58 (Emphasis mine)

A magic item’s description explains how the item works. Handling a magic item is enough to give a character a sense that something is extraordinary about the item. The identify spell is the fastest way to reveal an item’s properties. Alternatively, a character can focus on one magic item during a short rest, while being in physical contact with the item. At the end of the rest, the character learns the item’s properties, as well as how to use them. Potions are an exception; a little taste is enough to tell the taster what the potion does.

So, you can just taste potions to identify them. For all other items, you learn what they do immediately upon casting the identify spell, or you can spend a short rest identifying the item. The tasting is not drinking the whole potion down without knowing its effect. It's just a small sample, maybe a few drops or a small sip in quantity.

Flavor, such as what kind of experimentation you do to discover the properties, or how the properties manifest themselves to the characters, is up to DM discretion. Playstyles can vary from just telling the party what the items do after the identification, to giving the players clues and having them figure it out themselves.