Around 6 months ago my laptop bit the dust leaving me an entry level Acer I bought off of Amazon for around $300. That being said, I'm looking to do a little gaming in my free time and would love to spend some time with a game that has an excellent storyline. I love point and click adventure games and RPGs. My current graphics card is a Intel UHD 620.

Albion or the old infinity engine games (Planescape Torment, Baldur's Gate 2 etc.) should work.

This game is a really good RPG with an excellent storyline which might run fine on your system (so long as your system meets or exceeds the system requirements listed on the store page, of course): https://www.gog.com/game/the_legend_of_heroes_trails_in_the_sky

Divine Divinity, Beyond Divinity, Older Might and Magic, Forgotten Realms Archives, Avernum Series, Avadon, UnderRail, Arcanum, Planescape Torment (Original, and possibly enhanced edition), Fallout, Fallout 2, Geneforge, Baldur's Gate, Baldur's Gate 2, Albion, Septerra Core (maybe)

Adventure:
Day of the Tentacle, Maniac Mansion, Quest for Glory, King's Quest, Space Quest, Monkey Island, Sam and Max, Blackwell series, possibly A New Beginning: Final Cut...

Intel UHD 620 is not that bad, you can play all of pre-2008 games, not only old titles like Arcanum or RPGs from Spiderweb. that can also handle Morrowind, Oblivion, Fallout 3 and New Vegas and possibly Fallout 4 (if you have alot of RAM to cover video card shared memory). Basically it is safe to assume that this card can handle every game which requires under 1Gb VRAM, most games till 2008-2010 required only 512Mb of VRAM or less.

Dex

Cyberpunk RPG slash platformer slash beat up slash adventure

Baldur's Gate series, Icewind Dale series, Planescape Torment, Fallout Series, Arcanum, Divine Divinity, Wadjet Eye games (Gemini Rue, Resonance, Blackwell, Technobabylon, Primordia, Shardlight etc), Shadowrun series, Neverwinter Nights, The Witcher and many more. GOG has an abundance of great games for lower-end systems.

cRPG Fallout2
Point&Click: Gabriel Knight 1

Gemini Rue

That's easy: Fallout & Fallout 2 (especially). You won't need anything else to play for the next 3 months. You're welcome. ;)

I played KOTOR 2 recently on intel hd620 without problems. You can add it and KOTOR to the list

I played KOTOR 2 many years ago on some cheap radeon HD card with 256Mb of VRAM so it should fly on modern Intel chipsets. That's awesome game.

i would add to the list:
Deus Ex 1/2/3, Thief (all classic games and The Thief), Dungeon Siege 1/2, Sacred 1/2 or Titan Quest (these are more action RPG like Diablo), Bard's Tale (humor).
I agree about Arcanum, Spiderweb games and Beamdog games as well (Baldur's Gate, Icewind Dale, Neverwinter Nights 1/2, Planescape Tornment), Arcanum (one of a kind for me, very atmospheric game).

Point & click: I second Monkey Island and Gemini Rue and I add Indiana Jones and the Fate of Atlantis (great characters, gorgeous locations, three different paths) and Flight of the Amazon Queen (plain fun and it's free).

Heroine's Quest the Herald of Ragnarok!