How can player characters see into the Ethereal Plane? I'm familiar with the spell Truesight, but I'm wondering if there are other ways.
See into the Ethereal Plane yourself
The Ethereal Sight mode of Divination Wizards' 10th level feature The Third Eye grants the ability to see into the Ethereal Plane as an option.
Two spells explicitly lets a character see into the Ethereal Plane:
Then of course you can gain truesight, which per the description in the Monster Manual (page 9; or PHB p. 185):
[...] Furthermore, the [monster/creature] can see into the Ethereal plane within the same range.
Three magic items grant truesight as part of their effect (I'm not going to list magic items from modules as that would make the list essentially unbounded):
A Gem of Seeing (rare, requires attunement) will grant truesight for 10 minutes for each of its 3 charges.
A Crystal Ball of True Seeing (legendary, requires attunement; a crystal ball variant) adds truesight to the scrying spell cast through it.
The Eye of Veccna (artefact, requires attunement) grants truesight (no range listed) as one of its many benefits. No downsides, promise.
If applicable to your game, the Third Eye Nomad Dicipline from the UA Mystic Classgrants truesight. Alternately, there are monsters with it and spells that let PCs take the form of a monster. As there are no beasts with truesight (bummer, right?) we need to be able to take any form. True polymorph and shapechange are good bets and you can find a list of monsters with truesight here.
Travel into the Ethereal Plane
If you are on the Ethereal Plane, you can see other creatures on it (and interact with them generally).
Etherealness (7th level; Range: Self). This spell transports you to the Ethereal Plane and if cast at a higher level will let you bring some willing creatures along. (Horizon Walker Rangers get the ability to cast a limited version of this spell at 7th level.)
Blink (3rd level; Range: Self). This spell will transport you to the Ethereal Plane, but with the side effect of transporting you back at the start of your turn. It's a bit weird, but it works.
"Companions" to see into the Ethereal plane
If you don't need to see into the Ethereal yourself, but can be sated by a companion (i.e. another creature under your command/control) doing it, there are a few ways for that too.
Mordenkainen's faithful hound (4th level). The phantom watchdog created by this spell can see into the Ethereal plane and alert you of any activity there.
Conjure celestial (7th level). When cast at 7th level, this spell can summon a Couatl (CR 4), which has truesight; when cast at 9th, it can summon a Battleforce Angel or a Felidar, both with truesight and from the Guildmaster's Guide to Ravnica (ask your DM about availability).
Summon greater demon (4th). When cast at a higher level, this spell will let you summon a demon of CR one higher that spell level. Notably, Glabrezu and Master of Cruelties (again from Ravnica; ask your DM) are CR 9 demons with truesight and so can be summoned with a 8th level casting.