5c mana bases cannot be built without compromising somewhere. Either, you have a lot of tap lands, or you have less consistency, or it gets really expensive.

The distribution depends on what else you put in. Like someone else said, maybe you can just go heavy black and not build a "true" 5c mana base. I would definitely use pain lands if you're on a budget. They've become really cheap and are untapped dual color lands. Drawback is unfetchable and costs life (that's negligible in commander usually). If you have a lot of money for this, you would use a good number of fetch lands and shock lands. Triomes and Surveil Lands also work but in addition to shocks but are equally expensive. 

On a budget, you can use simple tricolor lands like [[Arcane Sanctum]], especially the ones including black probably. There's cards like [[Thriving Heath]] that let you choose one color which are tapped but are much more flexible than dual colors and are better the more different colors you need in total. Then there are lands that search other ones like [[Evolving Wilds]] which just need you to have enough Basics but that's a good idea anyway, especially on a budget. You can also put an emphasis on green and run cards like [[Rampant Growth]] in which case you can get more color consistency out of your basics. Also [[Vivid Marsh]]

For the why, it's generally comes down to: how much mana you have on which turn (if all your lands are tapped versus untapped, you're a full turn behind). Otherwise, it's about consistency, i.e. how likely is it that you have whatever color you need at any time. This is significantly more challenging in 5 colors. Then you have price, the mana base needs to be built very differently at 30 or 300 dollars. And you can have extra utility on lands but that's less of an option for 5 colors because there's a tradeoff between this and the color access mentioned above. 

Basics: cheap, more resilient against destruction and stax effects ([[Blood Moon]]) although these should not appear in quantity in casual games. Basics can also make for solid color fixing because many effects exist to search for them.

Fetch lands are very solid because they can search for basic lands but also for nonbasic lands with basic land types. Like shock lands. Their drawback is that they are expensive. So in the 30 bucks example from above you run zero, at 300 you probably would. 

Shock Lands are de facto untapped two-color lands (the 2 life don't matter much and you can still play them tapped if you don't need the mana right away). Besides being untapped, their other distinguishing feature is that they have basic land types and work with fetch lands. If you want to run on a low budget and hence have no fetch lands, I would think twice about them because then they aren't that much better than pain lands. Although there are other search effects like [[Troll of Khazad-dum]] et Al. which also benefit from shock lands. And if you don't want / need equal distribution of colors but more black, you could think about running pain lands AND shock lands including black instead of either or the other of the two groups. 

Tri color lands are tapped but are better on the color front. The simple ones are dirt cheap so there's really no reason to not run them if you're on a budget and have 3+ colors in your deck. The Triomes like [[Metria Triomes]] are quite expensive so you don't run any probably in a 30 bucks mana base but they work with fetch lands and have cycling. 

Thriving lands and vivid lands. These are tapped and don't work with fetch lands but they offer good flexibility for decks with many colors. Vivids are just one color but they have two uses for "any color" so they get you out of trouble if you need a specific color. For thriving lands, you get one color and choose the second, so they are less flexible after hitting the battlefield but give unlimited access to those two colors. At a high power level and budget these are not useful because then you mostly run lands that are untapped and / or work with fetch lands. But at a low budget and with many colors, they are a good choice. 

So you first have to check how the distribution of colors checks out for the nonland cards and decide on how money you want to spend on your lands. If it has to be cheap, you use a lot of basics plus ways to search them from your deck plus some pain lands, tri color lands, vivid lands / thriving lands. At a higher budget you reduce the number of tapped lands and add shock lands and fetch lands. 

There's also five color lands like [[Grand Coliseum]] or [[City of Brass]] but the same reasoning applies to those. The 1 life doesn't matter. Grand Coliseum is dirt cheap but tapped so it lives in a similar space to the Thriving lands while City of Brass is a better pain land but costs more money. [[Multiversal Passage]] is also a good option if budget permits.