l think something is wrong with our swimming pools filtering system.
We have not been able to set the pool up because of the price of filling it and maintaining it, but the last time we had it going, we noticed a problem with it keeping clean. l know nothing about the maintenance of a pool but could you give me advice on keeping one clean? And if there is something wrong with the filtering system...like if its broken or dirty or whatever, is there a way to check it and fix it?

You do not mention the type of filter. DE? Cartridge? Sand?
You also do not mention if ur water was properly balanced. If it was not then ur filter may not be the culprit.You filter is just one part of keeping a pool clean. It & everything else l will mention have to be up to snuff too. If u mention ur filter type in comments, l will check back later & edit my answer to reflect ur particular filter type & the things to do for it or things to check for.
Tips?
The secret to keeping a pool looking good & healthy to swim in is actually pretty simple.
1/ Filtration: Pool filtration equipment is sized specifically for ur pool is volume when it is installed. It is sized to turn over ur pool is water 1.5 times in a 24 hour period. If u use a timer on ur pump, u need to have a run time of 18 hours per day minimum. That will give u a turn over of 1.0. All of the pool is water gets through the filter & sanitizer ( chlorine dispenser/ salt generator/ ozonator/ or brominator) once in a 24 hour period. Anything less than 18 hours & not all the water is filtered & sanitized, leading to cloudy water eventually & higher chemical costs. Keep ur filter clean to maximize it is rate of flow. The more water going through it easier, the less ur electrical & chemical costs will be because the filter is working to it is best ability. You do not mention ur filter type so l can not be more specific.
2/ Water balance: You need to check ur chemistry at least once a week & adjust it as required. Typically, u will want to see a Total Alkalinity of 80 -120, a pH of 7.2- 7.6 & a Cl of 1.0 -1.5.. Do not ignore the TA. It is actually the most important thing to keep balanced. If it gets low, it can damage pool equipment & pool bottom surfaces. If it gets high, it can cause scaling. For either high or low conditions, it can also cause ur other readings to go out of whack too. Think of it as the governor on a car engine. It is the boss of all the other chemistry of ur pool. If it is set right, ur pH & Cl wo not have wild swings & the water is easier to control, saving u money in chemical costs.
3/ Sanitation: Keep the pool clean. Organics like leaves & twigs suck up chlorine & spit it out as useless, already used up chloramines. Fill up ur sanitizer at least once a week & make sure it is output will give u a constant Cl reading of 1.0 - 1.5. That is the safe level for swimming, such that the chlorine has little effect on u but does a lot of killing off of bacteria & algae. It wo not get all of it though. It is too low a dose to get it all. It will be comfortable for swimming while affording some protection. Do not skimp on the pucks & buy the ''unstabilized'' variety. Read the label & make sure it says ''stabilized''. It is important since there is a chemical called cyanuric acid that is added to stabilized pucks that can affect how ur chlorine deals with sunlight. Unstabilized chlorine gets zapped out of a pool pretty quickly, sometimes in an hour. It is not worth the trade off in price difference since u will use more of it.
When ur water is balanced properly & filtered properly, u still need to dose the pool with shock, usually once a week to get the organics that ur normal sanitizer missed. That missed stuff begins to add up & will eventually overcome the filter is & sanitizer is ability to deal with it. Your pool WILL go cloudy. Your water chemistry WILL go to hell.
If u keep ur TA,pH & Cl levels properly & ur filter working to it is max, u wo not need much shock, sometimes, depending on the weather, bather load ( amount of swimmers & pool use) & fill water, u may even be able to skip a week, especially if ur CL is a little elevated because no one used the pool & it is been cloudy, not windy & not raining since ur last dose. Shock is a stronger, more concentrated form of chlorine that is unstabilized. It goes to work quickly to nail any organics in the pool then after awhile it gets zapped by the sun & drops ur levels back to normal. Do not swim in a pool right after a shock. Not nice. Wait for the levels to drop. It may take 3 hours , it may take a day, so plan ur maintenance for a time when no one will be wanting to swim. Most people do it on a Sunday evening. Most swimming is done for the weekend & since it is evening or dusk, that yellow ball in the sky wo not knock ur chlorine levels down so much & u can actually save money by using less shock. Next morning the sun will hit it & in a short time, you're back in business for the week.
l know this all sounds like a lot, but when u actually do it, u will find that it takes no more than 45 minutes per week & that includes vacc'ing the pool. The flip side is that if u do not do it properly, u will be spending more time & money trying to fix it & u wo not be doing any swimming either.
With what you've given me to work with, l am afraid that is all l can suggest.

ScubaBob is once again right on, as a Swimming pool Tech myself, l would add the only other thing to do would be take a water sample to ur local pool store & have them test it for everything.

They will test everything possible for ur pool water. The computer will analyze the readings, & based on the information u give them, it will generate a list of chemicals & things for u to do in order to balance ur pool water properly. Balancing ur pool water properly in the beginning of the year WILL DEFINATELY save u money over the course of a year by not having to use extra chlorine or other chemicals trying to keep it clean.

l can not tell u how many times l have walked into a back yard, & then in talking with the customer they have told me that they r shocking the pool every other day just to keep it clean. After taking a water sample, & seeing that just about everything else was at improper levels, once they were corrected it was alot easier to maintain.