No
Haste says:
That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
Casting a spell or using a magic item fall outside of those options. The rules on activating magic items say the following:
If an item requires an action to activate, that action isn’t a function of the Use an Item action, so a feature such as the rogue’s Fast Hands can’t be used to activate the item.
Additionally, it seems you have a misconception about the spellcasting rules. The rule that limits you from casting other spells in a turn only applies when casting spells with a bonus action casting time. For example, an Eldritch Knight could use Action Surge to cast fireball and magic missile in the same turn.
A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven’t already taken a bonus action this turn. You can’t cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
When I had a new player start playing with us about 6 months ago, she had this exact problem. My solution was two-fold.
Reduce the number of options
She wanted to play a druid, so there were lots of spells to pick from. So I reduced the spell list from 7 cantrips and 16 level 1 spells to a list of 3 cantrips and 6 spells (1st level druids pick 2 and 4 respectively).
In my case, she didn't know what sort of role she wanted to fill, or what sort of things she wanted out of her spells, so I picked a decent range of generally useful spells. If your player(s) have a better idea (healing, DPR, charm, illusion, utility, etc...) then you can tailor the selection to that.
I highly suggest if doing this for a caster with semi-permanent spell selection (Bard, Eldritch Knight, Ranger, Arcane Trickster, Sorcerer, Warlock, Wizard) that you allow them to change out their spells more frequently than is permitted in the books as they learn what they do and don't like. For example, if your Bard finds out they aren't a fan of Charm Person, let them swap it out for free, even if they haven't just leveled up.
Make short descriptions of the spells
I also did this for my new player, as druids frequently change out spells. I wrote up short descriptions of each spell, and their general purpose. Some examples:
Shillelagh - hit things better
Entangle - make people move slower in an area
Faerie Fire - turn off invisibility and make people easier to hit
Thunderwave - AoE damage right next to you, and push away
This way, she had a quick idea of what each spell does, and could make the selection from there.
Final Notes
This method seemed to work well, although it's a sample size of 1, so YMMV. I think it requires a certain amount of trust between player and GM (since the GM is reducing their choices), but it gets them on their feet quickly, and if you allow them to change after the fact, it doesn't penalize them for their choices, which will make it easier on them to make a choice.
Best Answer
Going by the rules it could replace washing clothes (but not bathing), but these aren't features supported by rules in the first place.
Before I delve into what Prestidigitation can and can't do I should point out that, like Cubic said, there are no hard requirements in D&D 5e to wash yourself or your clothes. This is purely a thematic and narrative action that isn't supposed by the ruleset.
That said, let's look at what Prestidigitation does:
Whether you can clean your clothes or not depends on whether they're a cubic foot or less, which would ultimately be at your GM's discretion, as would whether you can clean them all at once or have to use the effect on each article of clothing individually.
However, it can't replace bathing, because player characters aren't objects; they're classed as creatures. No effect of the spell cleans a creature.
In short, the bathing going by RAW is a hard no because the spell can't do it, and the clothes question would be up to your GM's discretion.