That depends on the kind of pain you feel. There's a big difference between worn out achey muscles (which means you need rest) and a tear or chronic pain caused by bad fit. There are specific places where any pain is Very Bad, your joints and your back.
Anyway, yes, it's absolutely necessary to take recovery days and full rest days. A lot of beginners make every day a hammerfest and hurt themselves, burning themselves out on cycling. After a big effort, the next day should be a recovery day, where you do a light ride or walk, just moving around but not getting your heart rate up, just to get blood circulating into your poor tired muscles to help their healing process. Most days should be moderate efforts, especially when starting out, because you're building up your base endurance. Mixing in other exercises is also necessary. For instance, building up your core strength is huge, but doing so via cycling is ineffective, so some of your workout days should be core days with some upper body.
Usually my routine is a couple of workout rides a week, doing hill reps and stuff for around 20 miles, a "just ride" day, and a more intense workout ride. About twice a month I have a big day where I do an idiotic amount of climbing or something somewhere out in the wilderness. I also ride to work every day except for when the weather is impossibly shitty.
My "just ride" days are meandering around town at a reasonable pace on my mobile couch, from places I want to check out or maybe that one pizza shop with the killer slice or from park to park or something. I'll wind up doing 30 miles just farting around like that. It's "junk miles" but it's also why I started enjoying cycling in the first place. >>1391570
Maybe you could 3D print lugs like Prova and some others do? Or you could become a lug god like Darrel McCulloch and Mark Dinucci and hand-cut everything from raw steel.