It's cute that people still think an addon like HealBot, Vuhdo, Grid etc. plays the game for you.
No, it doesn't. It let's you costumize the UI in ways that the game either doesn't allow with it's base functions or allows you to quickly set keybinds without writing a macro for it. Macros have the benefit of working without the addon being up to date but HealBot and similiar healer UI addons offer far more option than that.
But healing is still done by the person, not the addon.

Back to topic:
Since you're just about to start healing, you'll go through the leveling phase first anyway. Damage in leveling dungeons, except a few infamous ones, is fairly low and you quickly get the feeling for damage patterns. On trash pulls, the tank usually is the only target that needs healing while on boss fights, it's usually the group that needs more healing.
You don't have back-to-back mechanics on leveling dungeon bosses, so after one mechanic went off that hit the whole party, you have at least several GCDs until you may need to heal. Since party heals are ususually either centered around you or ground target, you don't even need to switch target to use them.
Setting the boss to focus target helps because you can both see the castbar and predict incoming tankbusters (that usually tickle btw) or party-wides and you can assign a hotkey to target the focus target to quickly switch between tank and boss.
Mouseover macros are okay as a crutch but be aware that they don't queue so it will only go off if you use it right as the animation of the previous action finished. Meaning you have to use it until the cast starts before using a skill that does queue, otherwise the latter will take priority and you lose the macro heal.
E.g.:
Stone -> used macro'd Solace during cast -> pressed Stone right after = 2x Stone instead of Stone -> Solace -> Stone because you used the macro too early.
15 line macros slightly help alleviate the issue but you will nonetheless run into it and have macros not go off because you used them too early and they don't queue.

For party list costumization the options are unfortunately extremely limited. If the myriad of buffs that plop up during burst from some classes confuse you, can either limit of turn off showing buffs. Debuffs will always be displayed first and the dispellable ones will have a glowing white-blue line displayed above the debuff.
But you'll have to practice getting used to it because aside from scaling, placing, sorting and a little bit of buff/ debuff display costumization, there isn't a lot of useful features.

All in all, with how slow, low and easily predictable incoming damage here is, you'll probably get used to it fairly quickly.