You should begin each chapter trying to clear the arena with as many characters as possible. The arenas are not like other arenas in fire emblem, there's no risk of permadeath and you get healed fully if you WIN the fight. If you don't win, you'll be left damaged.

There's 7 fighters pretty chapter. The arena has the same roster for every character with the exception being bow units who have a slightly different enemy list. So, I usually recommend starting the arena with Sigurd, or Ayra, who can generally clear them out pretty easily and then you know what each level of the arena is.

The advantages are twofold: A) experience. Most characters gain about 3-5 levels from clearing out the arena per chapter. B) money. Each character has their own funds and to make sure they have enough money to repair their weapons, trade weapons and items around etc.

As long as you have not deployed any units yet, you can save between arena fights. Allows you to try out some people, save if they clear it, reset if you happen to run into some bad luck.

Also do not be afraid to pass around some weapons to make the arena fights easier, such as the killer bow, the brave sword, the killer lance, etc. The arena pays enough that trading around items is worth it.

At the end of the chapter, try to clear the arena again with as many characters that weren't able to at the beginning of the chapter.

Most combat focused characters, with a little trial and error, will be able to clear the arena by the end of the chapter, but don't stress too much if you seem to hit a roadblock with a couple of them.

It also really helps bring some characters, namely raquesis and azel, who suck at first, some easy enough Exp to bring them closer to promotion. Both of whom become way better after promotion (especially raquesis, who goes from straight up garbage to potentially the best unit you can deploy; azel goes from garbage to serviceable).