Good for you for asking for help!
Everyone has a slightly different method for sweeping and mopping. Keep at it, and you'll find one that works for you. Some of the things that make a difference include:
For sweeping, is the dirt mostly fluffy, such as pet hair or dust bunnies? If so you might find it easier to use a dust mop or cannister vacuum, if available. If you have to sweep fluffy dirt with a broom, do it more slowy and gently, to avoid kicking the dirt up in the air.
Type of bucket: first of all, you don't NEED a bucket if you can use the kitchen sink to hold the mop water, but that might gross out the other residents. I have a "split sink" and use it to hold the mop water whenever I'm working near the sink. Do you have a bucket with wheels and a wringer, like janitors use, or a plain ol' bucket? I have even used my kitchen garbage can as a mop bucket--the can is sparkling clean afterward!
Whatever bucket you have, don't fill it so full of water that it's too difficult for you to move around and empty. I change the mop water whenever it looks icky (by "icky" I mean I wouldn't want to put my hand in it,) so I don't fill my bucket very full--less than 10 inches of water is plenty. Even better is to use 2 buckets--one for the soapy water, and one to rinse the mop. This keeps the soapy water clean longer. When I pour out the water, I usually pour it down the toilet, especially if it's really dirty. This also cleans the toilet bowl, and is less likely to clog the plumbing than using the sink, since the toilet drain has a larger pipe. The more often you change the mop water, the cleaner your floor will be, since you're removing the dirt, instead of "rearranging" it.
Type of mop: sponge mops usually have some sort of device to squeeze out the sponge--either by pressing on the sponge, or rolling it through a wringer. Some "string" mops (either real string or narrow strips of fabric) also have a self-wringing feature. My Libman Wonder mop and the "Smart Mop" (as seen on TV) have a "sleeve" that you push down over the "strings" and twist to wring them out. Mops that don't have a self-wringer are usually paired with a bucket that has a wringer.
The type of flooring also makes a difference. Wood floors need different treatment than vinyl or linoleum. Flooring with a small pattern will not show the dirt as much as flooring that has a solid color. A very dirty floor might require more soap in the mop water, and/or rinsing the floor with the mop; if the floor is only lightly soiled, you can use less soap in the water, and not rinse it off. Most people use way too much soap.
To sweep, move as much furniture out of the way as you reasonably can. I start in one corner of the room, grasp the broom gently, with hands about 12" apart. By "gently," I mean just enough force to keep from dropping the broom. I start at the edge of the room, reaching as far under furniture as I can (dust bunnies and pet hair tend to gather around the perimeter of a room. I stand in one place, and put the broom bristles on the floor, as far to my left as I can (you might find it easier to put it on your right side.) I then slide the broom bristles towards me by pulling the handle slowly and gently (most of my floor dirt consists of fluffy dog hair.) Once the dirt is in front of me, I reach the broom out again, overlapping my previous stroke by 25-50%. I'm pulling the dirt towards me, rather than pushing it away from me. I repeat this process untill I've swept everything I can reach without taking a step. I then take 1 or 2 steps, and repeat the process.
After repeating this process 2 or 3 times, I use the dustpan. I do not try to sweep everything into one pile first, because I find it easier this way. I place the trash can next to me, press the edge of the dustpan firmly against the floor (I have to squat or bend over to do this,) and use either the broom or a wisk broom to sweep the dirt into the dustpan. If a little dirt remains at the edge of the dustpan, I move the edge back about 1," and sweep again. I then deposit the dirt in the trash can. I repeat the process until the whole pile is in the trash.
I repeat the process of gathering the dirt towards me, putting it in the dustpan, and emptying it into the trash can, until I have worked my way around the room.
If I have trouble getting all the dirt into the dustpan, I wet the edge of a piece of newspaper, stick it on the floor, and use that edge as a "dustpan," sweeping the dirt onto the newspaper.
When I mop, I fill my bucket (a cat litter bucket,) about 2/3 full of warm--hot water (not so full that I can't lift it easily.) I add about 2 tablespoons of dish soap or liquid hand soap. If I'm mopping my white vinyl floor, I also add a few tablespoons of chlorine bleach. If you prefer, you can use a product made especially for floor, such as spic n' span or lysol, but liquid soap works surprisingly well. I use the mop to stir the soap into the water, dip the mop until its head is wet, then wring it out as much water as I can. If the mop drips, I wring it out again.
I grasp the mop in the same place as the broom, but a bit more tightly. Since I use a string-type mop, I press the head on the floor, and work it in a "figure 8" pattern, while standing in one place. I go over the same place 2 or 3 times, until the floor looks clean. I then take a step BACKWARDS, and work the same figure 8 patter, overlapping the previous area slightly.
Unless the floor is very clean to begin with, I usually use a 2nd bucket of plain warm water to rinse the mop head, before dipping it back in the soapy water. Depending on how dirty the floor is, every 2-3 passes I dunk the dirty mop in the rinse water 2-3 times, wring it out, then dunk it in the soapy water, wring it out, and repeat the process. I change either bucket of water when it looks icky.
Depending on how quickly I need the floor to dry, I might turn on my box fan and direct it across the freshly mopped area.
If the floor is very dirty, I use more soap in the water. I then rinse out the mop in the plain water, and go over the soapy area with the mop and the plain water until I have picked up as much of the soap as possible. I try to avoid using more soap, because I never seem to get enough off the floor to avoid leaving a soapy film. If you have 2 mops, you can use one just for the rinse water, which helps a bit.
One of the keys is to start in the far corner of the room, and work your way backwards, so you don't have to step on the wet part of the floor. If you can't stay off the floor until it's completely dry, then mop 1/2 of the floor where you need to walk, allow it to dry, then mop the other half.
If there's dirt stuck to the floor, put a little soapy water on it, allow it to sit for a few minutes, then try mopping it up. If that doesn't work, add more soapy water, and use an old toothbrush, windshield scraper, or spatula to scrape it up. Don't use anything sharp.
If you're using a sponge mop, the main difference is that you work the mop forward and back, instead of in a figure 8.
A couple of alternatives to using a bucket of soapy water include:
Use a squirt bottle of soapy water or water and cleaner, and a bucket of plain water. Dip the mop in the water and wring, then squirt the soapy water on the floor. Work the mop in the same pattern as before.
Use a spray bottle with soapy water or water and cleaner, and a bucket of plain water. Spray the area to be mopped, allow it to sit for a few minutes. Dip the mop in the water, wring it out, and work the mop in the same pattern as before.
I use these techniques when I'm cleaning a small area, but you can clean an entire floor this way, if you prefer.
I hope some of this is helpful, and I'm so glad you chose a sober living house. They will have your best interest at heart, and encourage you to grow and develop new skills.