Tensei Brave how to level up characters runs on wins, not on a menu: a character grows from the experience they earn by fighting, and the skills they use in battle unlock as that character passes the levels that grant them.
Tensei Brave how to level up characters: experience is per character
Growth is per character rather than shared across the party, so two members can sit at the same level and still be a different distance from the next one. The developer corrected Ruka's growth curve and refreshed the experience table in the Version 1.0.9.6.5 update of 2024/04/08, which is why exact level-to-next distances are not identical for everyone.
Alongside the level, the game tracks a separate affinity (好感度) for each character. It moves up and down with the level, and the companions you travel with develop faster than the ones who are not in your party — so who you keep close changes how quickly they grow.
How skills unlock
Skills are unlocked by levelling a character up: they become available once that character passes the level which grants them. If a skill list will not open after an update, the fix is to level that character up again rather than to look for a menu that is not there.
Using skills in battle
Unlocked skills are fired from the スキル (Skill) command on the battle screen, which sits next to 攻撃 (Attack) and 防御 (Defend). Items can be used from the same command set, so consumables and abilities come out of the same menu. Choosing between them is a normal part of a turn — the combat guide walks through what the rest of the battle screen is showing while you decide.
Field abilities are not battle skills
Lighting up dark areas, breaking rocks and moving objects are field abilities, and they belong to the individual companion who has them, bound to the J / K / L / H keys on the field HUD. They are not the skills you unlock by levelling and they do not appear on the battle screen.
They arrive with the character rather than with her level, which is why the fastest way to get one is to keep recruiting. The recruit allies guide covers which heroine brings which ability and what happens when a rock refuses to break.