Goodbye and hello, as always:

I think Adric had a more realistic introduction and induction into the TARDIS crew. Unhappy with his life at that point and losing his brother, of course he stowed away with someone as incredible as the Doctor. For me Adric's behaviour was very believable as a companion. He had the best of intentions, but sometimes this wasn't appreciated.

for me the best intro was hex......cracking intro story with a bit of intrigue that was not really resolved until very late on, but his departure was one of the worst i am afraid he should have stayed.......

on TV i think martha's exit was unsatisfying, and clara's many entrances and exits became annoying.

i am afraid that when someone dies....they should stay dead

best TV intro i am not sure.....

Best intro: River Song as having a brilliant introduction. Could there be an argument for it being a departure as well?

Worst intro: Mel

Best departure: Romana. It might have felt sudden but it had been built up for a couple of stories and "She'll be superb" is a great closing line.

Worst departure: Dodo followed by Leela. Dodo because she was just ignored, Leela because it was a subplot that was barely introduced until the last episode of Invasion Of Time and was ridiculously out of character - frankly, if she was going to marry a guest character, I'd rather see her having arse-kicking adventures with Herrick on the way back to Minyos 2. Now that's an Unbound story I'd like to see!

The Harvest is a great little introduction to Hex. It falls under the same purview as The Face of Evil or Smith and Jones. Nice little self-contained tales that give us a glimpse into the companion's backstory as a means of getting them aboard the TARDIS. A Thousand Tiny Wings for Klein is one of my favourites for how evocative it all feels. The confrontation between O'Donnell and one of the Mau Mau is a highlight for showing how the former's fascism comes from a genuine nonunderstanding of how people work.

If it weren't for the retcon a couple episodes later, I'd have said that Peri's departure in Mindwarp is definitely up there. It crystallises in that moment how serious the Trial is and what the cost to the Doctor as a person has been. The look of horror and pain on his face has to be some of Colin Baker's finest acting and Michael Jayston is exquisite as the Valeyard twisting the knife in:
Same is true of Face the Raven. A really good companion departure... I just wish it'd been the only one.

She's actually got two. The other is a book by Gary Russell. Business Unusual ends with her sneaking aboard the TARDIS with the Doctor entirely aware she's done it, accepting the future that's come to pass and all it may eventually entail.

BEST HELLOS:
Polly & Ben
Harry Sullivan
Evelyn
Frobisher
Mary
Charley
Martha
Bill

Pretty much all of those companions have intro stories that 1) are good stories in their own right and 2) introduce the character very well and very efficiently and 3) make you want to see more of them in stories to come.

Weird exceptions to this are Susan, Ian and Barbara who all have good hellos AND goodbyes, although none of the three stories (the first episode notwithstanding) are all that amazing. But dang if Hartnell stories don't land an emotional moment, even when the story around them is less than stellar. Heck, even Katarina has a good emotional exit and she's barely a companion.

Oh, and on the topic of BEST GOODBYES:
Jo
Nyssa (in the audios)
Erimem
Charley (w/ Sixth Doctor)
Izzy

All wrapping up a solid arc. Jo stands out as a rare exception to "it always sucks when someone falls in love in a story and leaves because of it" because by gosh somehow it works and I'm not entirely sure why. Nyssa and Charley's audio arcs were a little drawn out for some people, but both endings really worked for me. Izzy could have gone on a little longer IMO, but I expect reading it in retrospect is different than reading her DWM stories for seven years or however long it was.

EDIT: Forgot the worst goodbyes and hellos! Dodo has a pretty weak intro (although Salvation made it better) but when you expand it out to novels, Sam Jones takes the cake. And goodbyes, again, yeah, Dodo probably wins but special points to Leela and Peri (although both departures lead to good stories set afterwards...)