I don't think there is one right solution for cleaning, you should experimant and see what works best for you... Look on the internet and you will see lots of different techniques.
I suggest you acquire the following items:
-Carbon Fibre Record Cleaning Brush
-Velvet Record Cleaning Brush
-Soft white lint-free cloth, preferably with a bit of texture
Old underpants would do quite nicely
-Isopropyl Alcohol (known as 'Rubbing Alcohol' in the USA
-Wax-free furniture polish
The best stuff I had was called 'Adritt', it's a Lidl brand... Right now I am using Mr. Sheen but it's not nearly as good as the Adritt was... What you are looking for is Aliphatic Hydrocarbons and Nonionic Surfacants / Anionic Surfacants. It must be wax-free. If it says it's OK for cleaning plastic (i.e. TVs and stereos) it should be fine for vinyl. If in doubt try on a low-value record, Engelburt Humperdink etc.!
Use the carbon fibre brush first to remove dust. I use furniture polish to clean very dirty records. Spray on the record or the brush. If you spray on the record turn the speed down to 33 so you get a smooth controlled spray. Turn the speed up to 45 and if you have pitch control turn it up to maximum, then work the polish in with the velvet brush. Don't use too much downward force, give the polish time and a few rotations to dissolve the dirt and grease without working it deep into the grooves. Use your finger on the centre label to push the record round with the force / torque necessary to really buff up the surface. The dirt and accumulated grime should now wipe off with a cloth quite easily. For lighter cleaning jobs or finishing off after this stage you can use Isoproyl Alcohol soaked into the velvet brush. A drop of detergent in the alcohol helps. Purists argue you should dilute the aclohol with distilled water and even add wetting agent (surfacants), I've never done it myself but I gather it does make a superior cleaning solution.
Alternative cleaning solutions include shower gel, I've also heard of baby soap being used. Never use alcohol to clean Shellac records (gramaphone records / 78s) - I learned that the hard way, it will discolour the playing surface. Use shower gel instead. Ultimately it's a question of experimenting and seeing what works best for you. To reiterate, don't try anything out on valuable or rare records until you've read a lot of stuff on the internet and tried some different techniques and solutions.
Never ever 'wet-play' records. Whatever cleaning solution you use should be dried off completely before you play the vinyl. Doing otherwise will work any residue deep into the grooves and really ruin the record. Again, don't make the same mistake I made... With a 1962 copy of 'Dream Baby' by 'Roy Orbison' too
Finally I will discuss a very extreme cleaning technique that should only be used as a last resort with very heavily soiled records. Use at your own risk.
I had a batch of 7" singles that had sat in a garage for years, some of the records had lain uncovered in a horizontal position and were extremely soiled with thick, solid, dusty grime. Nothing I tried would shift it and the vinyl was completely unplayable. In the end I placed each record on a tea towel (one by one I mean), and used a sponge kitchen scrubber of the type used to clean non-stick pans (Teflon) soaked in kitchen cleaner liquid to clean them in a circular motion... Once the solution has dissolved the dirt I used a clean cloth to wipe the surface clean. I think it helped that the scrubber was quite well-used so had probably lost a bit of it's abrasiveness... Amazingly the scrubber didn't scratch the vinyl at all... The ones I used were made from synthetic sponge with a white scouring pad on the top. Interestingly, I just saw a web-page that says Loofah is safe for non-stick pans, so perhaps that could be used. Again I emphasize use only as a last resort.
Hope these tips help, feel free to comment if you have any questions or anything to add