If you can't get through season one of TOS, you're not going to enjoy Trek.
It's rather arrogant to say that if they don't like season one of TOS, they're not going to enjoy Trek. There are tons of better seasons throughout the franchise out there in my opinion. It's really inappropriate to come out with a personal opinion as factual statement, especially when you're all but deterring a person away from the franchise as a whole.
Anyways... to adellone's OP...
Frankly there's not that much continuity or story arcs at all in TOS. It's just what it is. A smattering of relatively unconnected episodes, with the same cast, providing 45 minutes of sci-fi fun. If you're looking for some continuity with the TOS crew, I'd suggest watching movies 2-6.
Beyond that TNG is a slight improvement. While the main premise of the show is the Enterprise flying off to the random sci-fi encounter of the week, the characterization and growth is stronger and there are a number of mini-arcs that span the seven years.
DS9 is your best bet for continuity and great story arcs. Though that really doesn't pick up until season 3. The characters, both main and reoccurring guest stars, are great all featuring excellent development and growth and for the most part the there is a story to the whole show beyond just exploring the unknown. It's easily my favorite.
Voyager, beyond the theme of being lost in space trying to get home, really is just the random encounter of the week again, though there are some mini-arcs through the series. The characterization is there, and the characters do change, even if they're inconsistent at times. Overall the series is pretty good though.
Enterprise, well the first half is just them flying through space for the most part, quite often rehashing things the previous series had done. The second half gets some rather intense story arcs one of them lasting pretty much a whole season. The characters are probably the weakest of the franchise though.
Again, just my opinions adellone, if it helps you find a starting point. Feel free to watch and form your own. If you're looking to jump into the franchise with TOS, I'd recommend starting with the movies as I said earlier. It'll give you a chance to get to know the crew without the random episodic format as movies 2-6 are pretty much one continuous arc, each picking up after the last. Hope that helps.