Jogging in English can be either a noun and the -ing form of the verb to jog.
In French, it can only be a noun, the verb borrowed from English is jogger while more common translations are faire du jogging ou faire du footing (note that footing does not have the same meaning in French and in English).

In the first case, your sentence litterally means ‘I started doing the activity of jogging’ while in the second case it is more ‘I started to jog’ or ‘I started doing the action of jogging’.

Both could be translated the same way:

• J’ai commencé à faire du jogging
• J’ai commencé à faire du footing
• J’ai commencé à jogger (less common).
• J’ai commencé le jogging
• J’ai commencé mon jogging (this session in particular).

There are two important possibilities; the first is that in which the activity is being considered as a whole, and in which the start referred to is someone’s first experience with it before continuing with this activity on a more or less regular basis (I). The second is that in which it is a question only of a spell of the activity considered (II).

The verb "do" (both in English and French) might be used (1/ (used), 2/ (not used)). Usage depends on the particular noun. For instance, given the noun of activity "math", all possibilities are found. So, it can be said that there is a reason, which is not saying much as this reason is only that it is not idiomatic.

MATH I

• 1/ ♦ So I started doing math when I was very young. (ref.)
• 2/ ♦ For example, when Shanice started math, she also discovered that there were others who were weaker […] (ref.)

In French, the two possibilities are also correct.

• 1/ ♦ Ainsi, j’ai commencé à faire des mathématiques lorsque j’étais très jeune. (translation)
• 2/ ♦ Par exemple, lorsque Shanice a commencé les mathématiques elle a aussi découvert qu’il y en avait d’autres qui étaient plus faibles […]. (translation)

MATH II

• 1/ ♦ Debwa then spent several minutes denying this, saying the form was one sent to Fierce County. She again started doing math in her notebook. (ref.) ♦ If you started doing math right away, you might immediately jot down something like J = 3d + c, thereby making the problem a lot tougher by introducing a variable […] (ref.)
• 2/ ♦ It felt very good and official to open my science book flat on the desk and flip through the pages while I answered the questions using the Caroline Method. After I finished but before I started math, i looked at Dad’s books. (ref.)

In French you do not use "faire des mathématiques" in this case, but instead "calculer" or something like that.

JOGGING I

In English only the second possibility is correct.

• 1/ — ("started doing jogging" is not used.) (ref.)
• 2/ ♦ "When they told me I was going to be crippled I knew better," she added. "And once I got back on my feet, I felt so good that walking didn’t seem like such a big deal after all. "That’s when I started jogging. (ref.)

In French, the first possibility is correct, and the second is only marginally correct.

• 1/ ♦ Marc , tu pourras commencer à faire du jogging le long de la Seine pour te mettre en forme ; mieux , tu pourras faire de la bicyclette , t’entraîner pour 10 Tiercé (ref.)
• 2/ — ("Commencé le jogging" is not used.) (ref.)
  ♦ Bowerman montra surtout de l ‘ enthousiasme pour ce qu ‘ il avait appris sur le jogging . Uhrhammer qui par la suite se mit au jogging après avoir subi une opération à (ref.)
  However, "se mettre au jogging", which should be common enough, is used, and that legitimates "commencer le jogging" because the two terms are synonymous enough; nevertheless, "se mettre à" is more idiomatic.

JOGGING II

• 1/ — ("started doing jogging" is not used.) (ref.)
• 2/ ♦ The Kid climbed to safety and started jogging northward, cutting through the historic University of Tampa campus and the famous Tampa Prep (ref.)

In French, the name of the activity wouldn’t be used alone; instead the verb "jogger" is what is proper; this is true except if the name is modified.

• 1/ "Commencer le jogging" (where "le" can in no case be the generic article but has to be the determinative article) is not common; the name of the activity is usual when certain modifications can be applied to it.
  ♦ Le temps d’enfiler un short, un T-shirt à manches longues et ses chaussures de sport, puis elle attrapa ses clés et son portable et entama son jogging du matin (ref.)

• 2/ ♦ Laurie commence à jogger. Je lui emboite le pas. (ref)
  ♦ Elle pénétra dans le parc à la hauteur de la 72e Rue et commença son jogging en direction du sud. (ref.)

GYMNASTIQUE In this case both possibilities in "I" are fully idiomatic.

I

• 1/ commencé à faire de la gymnastique
• 2/ commencé la gymnastique

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In addition to Sacha’s answer, If you want to say that you started to go jogging as a hobby, you could use:

Je me suis mis(e) au jogging

Or

Je me suis mis(e) à la course à pied

"Course à pied" has the advantage not being an anglicism, but is theoritically more generic than "jogging" as it covers eventual training for running short distances whereas jogging implies longer ones, at least in my mind.