Since this question is asked a lot, I figured I would make a quick little guide on how to make bits, compiled together of all the answers contributed from the various help posts asking this question. At least until a proper guide is made and the wiki is updated.

First rule of advice is stick with a low amount of horses when first starting out. Having more horses than you can afford will sink your money fast. Have no more than you can board at home because boarding fees can get expensive. As you get more money, focus on expanding your estate and getting more stalls to house more horses before you buy them. You can have up to 150 acres and 100 boarding passes as a basic account.

Buy cheap, decent horses. Check the rescue, sometimes you can find some good ones hiding in there. Look for at least a couple of hundred PS and younger horses to work with longer. If you can get little to no red in Breed Standard that can be helpful too but not necessary right away.

Focus on careers first. Horses can enter careers 5 times a day and it costs nothing to enter so you always make money. As they level up careers, they make more money.

Explore, explore, explore. In the dark horse mines you have chances to get gems, bits and ingots. Less likely of the latter but still a small possibility. You don't get many of the bits or ingots if you do find them, but gems can net you a couple thousand per gem depending on the type. You can also encrust tack as well and sell for a bit of a higher price.

The scrapyard could either be junk or hidden gems. You may find nothing, or basic tack, or you might find things like dyes, companions, or breed coins which could net you some money if you sell them under market trades.

Having horses specifically for exploring Hoofbeat Plains is a good idea. This is because they will use all their energy exploring so you cant enter careers with them. They dont need to be anything fancy either, any horse will do, just grab a cheap rescue and go. You can find all sorts of things in the explore map. Dyes, treats, basic flowers, or seasonal items. However, not only does exploring use your horses energy, picking up items uses your energy. Your energy is used in the mines, scrapyard, and picking up items in the explore. Unlike horses whose energy refills daily, yours refills throughout the day.

As a basic account, you can create 3 shows a day, six if you purchase more arenas as a basic account. Shows run after 24 hours and will net you some income.

Play Higher or Lower in the game room. You can play an unlimited amount of times and you earn 100 bits when you guess correctly. There is also now a new game, rock, paper, scissors. You can play 20 times and get $100 bits if you win. The game room can be accessed if you click Town in the menu. 

Once you have a steady income then you can focus on discipline and halter shows. Higher PS horses compete at higher levels and earn more money and all green BS horses do well in halter. Try to enter disciplines with less than 20 entrants instead of hundreds. I usually enter later runnings but I almost never have problems entering 10 shows under 20, occasionally under 50. Sometimes its hard with lower tier ones like foundation, local and state but try your best. If there are hundreds of entrants and you don't think you will be in the money, hold off and get your earnings a different way. You can enter 5 halter shows with horses with good breed standards.

When you are ready to breed, focus on quality. Purebreds or recognized crossbreeds. Wait a bit for campaigns, but random mutt crosses don't always sell well. Let your horses hit level 50 so foals can have at least a minimum training boost of 10. To find what a foals training boost will be, add the stud and broods levels and divide by 10. Everyone looks for different things in horses, high PS, high BS, levels, pretty colors, etc. As much as you may want to breed whatever horses you want, and you can, you might want to have popular breeds for profit. If you ever arent sure on interest or quality of your horses, always feel free to ask, especially in the discord as it is always active.

It is also probably a good idea to not start with breeding grades from scratch as it requires a lot of crossbreeding to get back to grade, which takes time and money. 

When you have good breeding stock, not only can you make money on selling their offspring, you can also offer them for stud and brood as an extra way to make some money.

If you get ingots either from various ways in the game or you have the financial means to buy them (this is not necessary though, just another bonus way to get money) the going rate is generally 350-500 bits per ingot.

These should generally help starting out with making money. You should keep some money in the banks as well because you earn a bit of interest every Sunday. 

Here are also some good wiki links to look at as well: 

Careers

Training and all of its sub tabs

Breeding and its sub tabs

Showing and its sub tabs

Hopefully this can be helpful to new players and cut down on the repeated questions, but please, feel free to add more advice here if you have any of ask questions if you dont understand something