For someone who hasn't seen either of the two shows, in order to get the most clear and continuous plot experience, does it make sense to watch Better Call Saul before Breaking Bad?
It doesn't matter.
Surprisingly, despite being set before Breaking Bad your enjoyment of Better Call Saul does not depend on whether you have seen Breaking Bad or not. Although characters from Breaking Bad make an appearance in Better Call Saul, as of the end of the first season you do not need to know anything about Breaking Bad to enjoy Better Call Saul, and you do not need to know anything about Better Call Saul to enjoy Breaking Bad.
However, there is no saying that this won't change in the future, and as such I recommend that you watch them in the order they were released — so, all seasons of Breaking Bad first, followed by Better Call Saul.
Edit: As of the end of the second season of Better Call Saul, I stand by this answer for the most part. The second season has reintroduced more characters from Breaking Bad, and although knowledge of them isn't required to understand what is happening in Better Call Saul, some of their appearances have more impact if you know who they are from Breaking Bad.
Edit 2: As of the end of the third season of Better Call Saul, I now think that the best way to enjoy Better Call Saul is by having first seen all of Breaking Bad. As the show has progressed, it has slowly reintroduced more and more characters from Breaking Bad and shown us events and interactions only hinted at in Breaking Bad - these are far more enjoyable with the context provided by Breaking Bad rather than relying solely on the information given to us by Better Call Saul. While Better Call Saul can still be watched and fully understand on its own, I now believe that those who have first seen Breaking Bad will enjoy the show significantly more.
Prequels are just sequels that happen to take place in-universe at a time prior to the original. In that it's a sequel at heart, there will be arguments to watch it after the original.
There's no hard-and-fast rules but typically a prequel will subtly reference the original. While it's not a requirement to see the original to understand the prequel, it's often a bit more enjoyable as you'll be able to spot those references.
I think the best analogy I can think of is Pulp Fiction. Could pulp fiction be re-edited so the entire film takes place in chronological order? Of course. Would it be as enjoyable of an experience? Probably not.
At this point it is hard to know, because Better Call Saul is mostly a prequel, but each season also opens with a scene (shot in black and white) close to the end of Breaking Bad. It's unclear if Better Call Saul will eventually recover parts of Breaking Bad and/or if later seasons will jump ahead and tell a full fledged story in the post Breaking Bad world.
“We couldn't really give you an exact amount of episodes," Gilligan told us. "The reason being—and I'm not being coy here—it’s just hard to know exactly. But you did put your finger on something important. Just from watching this show you can tell that it's a finite story. And we know that even further from the fact that this show has to butt up against the beginning of Breaking Bad. So there is a finite nature here.”
But there's one difference in Better Call Saul’s finite nature that wasn't there with Breaking Bad, which is that there is yet again the possibility of a whole other story to be told through the black-and-white beginnings of a post-Breaking Bad world that we've put at the top of each season," Gilligan says. "So while I think that there is a definite end in sight for the pre-Breaking Bad story, there still seems like there could be a lot in the post-Breaking Bad world. I'm kind of fascinated by that, simply as one of the first fans of the series. What could come out of that? No promises, but it seems to me that there's a little more opportunity for scope there than there even was in Breaking Bad.” http://www.denofgeek.com/us/tv/better-call-saul/264179/better-call-saul-could-become-breaking-bad-epilogue-vince-gilligan-reveals
Season 4 However, has been teased to go right up against Breaking Bad, including introducing more characters that were mentioned in Breaking Bad, but never before seen.
“This is the best season yet,” Vince Gilligan told Comic-Con today of Better Call Saul’s fourth season that starts next month. “The overlap between Better Call Saul and Breaking Bad gets closer and closer,” added the creator of the Bryan Cranston-Aaron Paul Emmy winner and co-creator of its prequel of sorts, driving the cavernous Hall H crazy with anticipation.
“You are going to see some stuff,” Gould cryptically declared before showing a clip of the Bob Odenkirk-portrayed slimy lawyer who first appeared on Breaking Bad in its second season in 2009. “You want the last hill on the rollercoaster to be the scariest of all,” Gilligan added. https://deadline.com/2018/07/better-call-saul-breaking-bad-crossover-tease-season-comic-con-panel-trailer-video-1202429718/
UPDATE - Future Scenes:
Upon the new "future scene" in episode 4.01's opening sequence, it has been suggested by Vince Gilligan, Peter Gould, and Bob Odenkirk that these future scenes may take place right before the end of Breaking Bad in between episodes 5.15 & 5.16. Also Episode 4.01 features an uncredited cameo from a well-known Breaking Bad character. Tonelly there is also a shift in mood, which is most likely closer to how Breaking Bad feels in comparison.
From Screen Rant discussing a recent Podcast: SPOILER Warning!!! The full article contains Breaking Bad spoilers AND discusses Better Call Saul 4.01 opening scene.
On this week's edition of the Better Call Saul Insider podcast (via The Wrap), a brand new, quite intriguing curveball has been thrown into the mix when it comes to those aforementioned flashfowards. While speaking on the program, Gilligan, Gould, and Odenkirk suggested that these sequences might be set prior to the events of Breaking Bad's series finale, and that the person "Gene" most fears the wrath of is possibly ...
UPDATE #2 - New Movie:
As it turns out there will be a feature film that takes place post Breaking Bad, featuring the character Jesse Pinkerman.
Not much is know about this yet, including if 'Breaking Bad' will be included in it's title or if it will get it's own name, but more over it doesn't also explain what this would mean for Better Call Saul's future either, if that character will end up in this film or if Jimmy McGill will still also go on past Breaking Bad on the TV series or if it will stay more strictly a prequel?
I'm someone who has not seen more than the Pilot of Breaking Bad, but I do know some spoilers/character arcs. I know that most likley Breaking Bad viewers are getting a lot more out of Better Call Saul then I am, because there are a lot of easter eggs and such, but I have decided to wait until Better Call Saul is over and have the opposite experience, as I enjoy the show greatly without knowing every little detail of Breaking Bad.
If Better Call Saul was never made (shudder the thought), then Breaking Bad could be enjoyed on it's own. I watched Breaking Bad first, and now am watching Better Call Saul. I think it's kind of cool to go back in time as "Saul's" character is built for the audience. Both shows can stand on their own, though having Both of them makes for a synchronicity that is of the highest level in my opinion.
You should watch the shows in the order they were released, not in chronological order of the scenes.
On a practical level, if you watched BCS first and became familiar with the characters, you would experience a jarring jolt to your suspension of disbelief when they noticeably lost a decade in age. Mike Ermentraut would blow your mind.
More importantly, it's a common trick in BCS for a character, familiar from BB, to appear in an apparently innocuous scene. However, we, having seen BB, recognise him immediately as a key character from that show (and therefore from the future of BCS). This gives us insider knowledge about his motivations and likely behaviour that is unknown to the other characters in the scene. Tuco Salamanca popping his head out the door is a good example of this. You would lose this effect if you watched BCS first.