Mortar: the Mortar menu's Check mods and Mortar for updates, or Settings › Updates (which can include betas). An update applies when you close Mortar, or choose Restart now. The Mortar menu's What is new in <version> shows what changed.
Mods: updates show on the mod list; Settings › Updates has Review mod updates.
Mod loaders and SMAPI: Mortar tells you when a new version is out; each game's settings hold the loader version and update it.