The whole point of cycling clothing is to offer greatly increased comfort over any normal clothing. Even if you look a little dorky. And, thing is, I can ride 20-30 miles in street clothes. So you don't even really need any of them until you are really trying to bike hard and fast and over longer distances.
Sometimes it's more important than others. For example, bike shorts are a huge contribution to comfort while cycling. There are a lot of variants on the design. Sometimes they hold themselves up with an elastic band around the waist. Sometimes they are "bib shorts" that have shoulder straps, which tends to work great for men and not quite as well for some women. They make skorts for women.
The big design point is that it must be comfortably form-fitting yet also not contain any seams in places that will bug your skin. There's a certain amount of individual fit involved. This means that you might like one brand and your buddy may like another brand and, were you to swap shorts (Don't do it! That's gross!) you'd really hate each other's shorts. Shorts are also generally gender-specific.
Now, if you want to bike in the cold, you can get tights that fit like bike shorts but cover the whole leg, or you can get leg-warmers that effectively do the same thing. The nice thing about leg warmers is that they can be taken off mid-ride. You can start the ride with leg-warmers and bike shorts and then take the leg-warmers off when it gets too hot. The downside of leg-warmers is that they sometimes will ride down. You can also get knee-warmers.
Pretty much, if you want to go long distances, you need bike shorts and maybe legwarmers or tights. That's my advice.
If you are really not wanting to look like a spandex cowboy, they do make "mountain bike" shorts. Basically, it's a pair of normal bike shorts with a matching over-short. So there's nothing preventing you from wearing a pair of normal shorts over bike shorts. Or a skirt. Or anything else. It ends up that most of the hardcore competition mountain bikers don't wear "mountain bike" shorts.
Oh, and one more thing. If a guy rides around in a pair of white shorts, he's going to be displaying a lot more than he realizes. Most men's shorts have at least a black panel to leave the crotch in shadow.
On the other hand, the rest of the ensemble is much less of a requirement than you'd think. I biked for a long time with my one pair of bike shorts and a cotton t-shirt. Eventually, because I needed them for riding in colder weather, I ended up with some long-sleeved wicking t-shirts.
I have an argument that isn't necessarily very popular, but I'll make it anyway. Sunscreen is not something you can blindly apply. There is a risk of harm from using it long-term that's fairly minimal, but not zero. Consider the traditional wardrobe from the hotter parts of India or the middle east. People don't wander around in minimalist attire, they are all covered up by light colored lightweight fabric. I think that, for rides when it's not necessarily humid, it's better to wear a lightweight long-sleeved top instead of a short-sleeved top. Which I haven't been able to find, but my long-sleeved wicking t-shirts worked perfectly well as.
On the other hand, I've got bike jerseys these days.
The big thing that bike-specific tops try to solve is air-and-moisture-flow. See, if you just rode up a mountain, you've been sweating a lot. If you then descend and have your jersey flapping in the breeze, it's pumping air very quickly past your skin. Which means that you just went from way-too-hot to way-too-cold. The shivers are a great way to have a hard time controlling your bike on a fast descent, mind you.
So, you want to use technical fabrics or wool, which I talked about earlier, to let water evaporate without getting wind or rain through. And you want to have the jersey form-fitting such that air can't cause the jersey to billow in and out. This means that people with too much belly fat look really silly in them.
They also like to help make the outfit more bike-ergonomic. Thus, they tend to have the back longer than the front, because you'll be hunched over and don't want to get your butt-crack sunburned. Pockets in the front are bad because it'll just bounce against your chest or legs while you ride, so they generally have pockets in the back for storing stuff like water bottles, pumps, tubes, phones, keys, etc. And you want a big long zipper down the front so you can unzip partway down while climbing for a little more airflow and then zip it all the way up while descending for a little less airflow.
In the heat of summertime, wardrobe is simple. A short sleeved jersey, maybe a rain shell in case it rains.
When the weather is cold or it's subject to change, it gets harder. Thus, the hardcore usually have a whole set of interlocking items depending on the weather, such that they can handle anything from summer heat down to freezing just by adding or subtracting items. You can get arm-warmers that cover your arms for when it gets a little cooler but can fit in a jersey pocket the rest of the time. Or you can get a full fledged long-sleeved jersey. Sometimes you need a vest to keep your core a little warmer, which are also often small enough to fit in a jersey. And there are rain shells that fold up very small, have breathing panels, and are see-through so you can still see sponsor logos.
If you join a club or team, there's usually branded jerseys that you can get, to show your allegiance. For a team, it's basically compulsory for you to be seen in your team kit. There's usually a team logo, but also room for sponsor logos. Back in the days of knit wool jerseys, they would need to embroider the logos. Later, synthetic materials let them do dye-diffusion transfers to a jersey, which means that sponsor logos are easily added.
You can also buy replica jerseys from the big teams, which make you slower and pretty much mean that everybody will make fun of you and your silly jersey behind your back. So don't do it.
You can get rain shells dedicated to cycling use. Now, you are probably asking yourself, "Why should I spend $150 on a rain shell for my bike when I can spend $30-50 for a normal sports-oriented rain shell?" Generally, the bike-oriented rain jacket will fit you better and have the pockets and zippers in the right places. Although I'm not actually sure how much of a real difference it makes, to be honest.
For a few areas, normal wardrobe with separate parts must be dispensed with. For time trials, you can be a little faster with an aerodynamic skinsuit. For cyclocross, there's so much lifting up and down of the bike that you really want a one-piece outfit as well so that it doesn't catch.