Look within yourself first and foremost. DESIGN what you are going to do ahead of time. THINK about how you generate the results you get. Your Excel tables ARE a starting place but because they are forced (by the nature of the beast that is Excel) to be flat files or darned near it.
In ANY non-trival Access application, you must decide what entities are part of this thing that you are tracking. You see, sometimes it helps to think of Access as a MODELING tool. A BUSINESS or DATAFLOW modeling tool. So step number 1 is to think about what it is you are modeling. From the brief description you gave, I firmly believe this CAN be done in Access and you will like the result, perhaps. But you won't like getting there because Access is so new to you that you are going to face issues of wrapping your mind around the path as opposed to the goal. (It is normal, trust me. Many of us who came from Excel origins go through the mind-shift to Access.)
OK, look for things that can be grouped together logically. Try to decide who or what naturally goes together, what COULD go together, and what things are so different from each other that it makes absolutely no sense for them to be together. You are identifying model entities. Obvious one: Horse. Another obvious one: A competition event.
I'm going out on a limb here, but you might be able to do this with two tables and a little faith. In Access it is OK to be slightly wasteful. Not MASSIVELY wasteful...
I see two or three tables and a couple of special queries.
Table #1 - Competitors
This will hold data about the horse. Use any identification scheme you want as the primary key, including a manually entered OR automatically generated entry number. Search this forum for articles on primary keys to read much more than you wanted to know on the topic.
IF persons can enter more than one horse or a horse can have more than one owner, and if you WANTED to track this level of detail, another table for Owners would be appropriate. But this would be an ADJUNCT table and it might even be optional.
Table #2 - Competitions
Here you put each event's results. This table would identify the competitor through what we call a foreign key - essentially a pointer back to the entry in the first table. The record would contain more data than is appropriate for some events, but it is OK to have blanks now and then. I'm thinking like this:
tblCompetitions
CompID - ID number of the competitor being recorded.
EventName - C, D, or J (for cross-country, dressage, or jumping)
Possible data for each event follows...
Score - whatever score is appropriate for the event. You will compute this via some update queries.
Placement (Placing?)
Jump Penalties
Time - but could be blank for dressage
Time Penalties
Other contributors to scoring.
Now, normally you do not store a computed value, but here you might violate the rule very slightly. 'cause it's not really a rule, it's more of a guideline. You would have some update queries that you would use to compute scores based on the data in each competition record, but the queries would be very selective. Since you have the event type in the record, each query could select for a single event and update the score based on record contents.
Then a simple aggregate query could be used to show the final score.
Search Access help for the terms I have used such as "Aggregate Query" and "Update Query" and any other that you don't recognize.
One more thing, and you might hate me for this... but if you are truly a beginner, this is productive in the long run. Google-search the web for keywords "DATABASE NORMALIZATION" and look at a few article from .EDU sites you recognize. Have this information in mind when you start to identify your entities.