Potential forms: ability versus not doing
Potential forms express ability or situational possibility. Distinguish not doing something from being unable to do it, and check the changes for each verb group.
Form the potential
For group 1, change the final u-row sound to the e-row and add る: 読む → 読める and 書く → 書ける. The polite forms are 読めます and 書けます. Notice the small difference between 読みます and 読めます.
For group 2, use 食べる → 食べられる and 見る → 見られる. する becomes できる and 来る becomes 来られる（こられる）. Forms such as 食べれる occur in conversation, but learn the standard forms as your starting point.
Ability and situational possibility
日本語が読めます。 can express the ability to read Japanese. この店で買えます。 expresses that something can be bought at this shop. Potential forms are not limited to a person's learned skills.
漢字を読む describes reading kanji; 漢字が読める expresses being able to read them. が is common with potential forms, but を is also used in some contexts, so do not treat every を as an error.
Negatives are not permissions
今日は行かない means 'I am not going today,' while 今日は行けない means 'I cannot go today.' These can distinguish a choice from an obstacle, so read the negative form carefully.
ここで写真が撮れますか。 can ask whether photography is possible. To ask explicitly for permission, use 写真を撮ってもいいですか。 Do not automatically equate practical possibility with having permission.
Check: What is the difference between 読まない and 読めない?
Show the answer and why
読まない means 'does not read'; 読めない means 'cannot read.' The latter expresses inability due to skill or circumstances.