Level up, do you mean via your player level from the quests/tutorial or level up horses yourself?

If you mean for player, if you are level 5 and 550/600, you are caught up with everyone else and at the moment, we are waiting on more quests to come along.

As for horses? Being a new player and putting them into Careers is the best way to go. You do this by going to the profile of the horse in question. Hit the little pencil icon to edit their page, scroll down a little and on right hand side, there with be a box that has 4 boxes to tick and two discipline drop down menus along with a Careers one. Your horses must be 2 years old and have their personalities completely filled out done via the vet. Tick the box for Careers only and leave the others unticked and you can send them off every 30 minutes. It is slow but they do earn levels and easy cash without needing to spend anything. 

It is slow in the beginning but it takes time, patience and practice to get better bred horses that can do better in Discplines once the stallion and mare are level 50, or any variation that equates to 100 as adding the two levels can give 100, higher or lower, divide that by 10 and you will get 10.0 for example. The 10 is the training boost your foal will have and it will do better training wise and in the show pen. Try seeing if there is any decent horses in the Rescue to adopt some cheaper horses. You can either put them into Careers or put them into Explore and go Explore Hoofbeat Plains to find the latest Christmas items to sell if you need cash that way.

It took me a little while to get the hang of things. The wiki helped some, but joining the discord really helped me the most. There's almost always someone around to help answer questions. I also learned a lot just by reading interactions between players and seeing what other players have been doing in the game. I remember being overwhelmed at first, and there is a bit of a learning curve, but once you get the hang of it I find it a lot more enjoyable and profitable as well.