I grew up with the Times crossword over dinner. We're not the most talkative family so a puzzle gives us something to do together while we're eating. A few observations, learned from introducing various other people to the hobby over the years:

1) The number one problem for new people is simply absorbing the rules of how to parse a clue. The linked article does a good job of going through these. Try to avoid the sense of learned hopelessness that often sets in early on.

2) The second most common isue is not managing in practice to get away from a literal reading. You need to take every word in isolation and try to escape the inevitable misdirection. For example, if the word 'rose' is in a clue that also contains the word 'flower', it's very unlikely that you're supposed to read 'flower' to mean something with petals. You need to think of any possible interpretation other than the obvious one. The classic second meaning here would be to read it as 'something that flows', which will mean the name of a river. Which brings us neatly onto the third difficulty:

3) Cryptic crosswords are heavily grounded in old-fashioned English culture. A reference to a river could mean a major international river or an obscure one from the British Isles, but it would be considered unfair to refer to an obscure river from another country. There are also some incredibly dusty references: 'sailor' could mean 'tar' (an archaic english slang that now exists only in crosswords) or 'AB' for 'able-bodied [seaman]'. 'Men' could refer to 'RA' (Royal Artillery) or 'RE' (Royal Engineers) among other things. One of the worst is 'posh' (or synonyms of it) to clue the letter 'u', which comes from high-society slang in the early-mid 20th century. 'Home counties' would be SE for South East [of England]. Cockney rhyming slang also often features, among many other things.

Problem 3 is the most insurmountable. It's also largely unfixable. If the range of acceptable references were broadened, it would become almost impossible for anyone to finish any given puzzle. But the references were fixed at a time that is no longer relevant and provides a huge barrier to entry for new people, which is why I expect these puzzles will largely die out over the next generation.

The bottom line is that if you feel like you struggle with cryptic crosswords, it's probably not because you're being stupid. There's a surprising amount of domain-specific knowledge you have to absorb, and in the best of cases any given puzzle will typically contain one or two absolute stinkers.
A few references to help:

1) A list of common abbreviations: https://www.dummies.com/games/crossword-puzzles/cryptic-cros...

2) A website that solves clues and tries to explain: https://www.crosswordgenius.com/

3) A blog where people solve puzzles and explain them so you can learn how it works: https://www.fifteensquared.net/

4) I haven't seen squarepursuit before (linked by tclancy) but it looks like an excellent resource.

A few random tips:

1) Try to get the clues from the first row and the first column early on, as these give you starting letters for other clues

2) Get used to looking for anagram indicators. 'drunk', 'rotten', 'altered', anything like that. The other anagram indicator is always from adding up letters: if the answer is nine letters, look for combinations of words that add up to nine letters. Once you identify them, anagrams are a solid place to start once you've looked at the first row/column.

3) If possible, crosswords are best done with company. Everyone thinks in different ways.

4) Be wary of fish references. These can be incredibly obscure, and often indicate that setter was struggling to clue the last few letters so just googled them and found some vietnamese river fish that fit the bill.

5) If you're truly fed up, you can use a thesaurus on the word that you think is the definition. This is a bad habit, but if it lets you open up the puzzle a bit then it might be the right course of action.

My favourite clue ever: 'geg' (9-3)