Los sábados juego ajedrez con mi abuelo en un tablero viejo.

Spanish usually expresses “on + day(s) of the week” with the definite article, not a preposition.

• El sábado = on Saturday (this coming/that specific Saturday, depending on context)
• Los sábados = on Saturdays (habitually, every Saturday)

You do not say en los sábados for this meaning. Just el / los + day is the normal pattern.

So:

• Los sábados juego ajedrez… = On Saturdays I play chess…

It’s plural because the action is habitual, happening every Saturday:

• El sábado juego ajedrez… = On Saturday I’m playing chess… (one Saturday)
• Los sábados juego ajedrez… = On Saturdays I play chess… (every Saturday, regularly)

About capitalization:

• In Spanish, days of the week are not capitalized in normal text: lunes, martes, miércoles, jueves, viernes, sábado, domingo.
• They’re only capitalized at the beginning of a sentence or in titles, just like any other common noun.

Jugar is the infinitive (to play).
Juego is the present tense, 1st person singular (I play):

• Infinitive: jugar (to play)
• Yo juego – I play
• Tú juegas – you play
• Él / Ella juega – he/she plays
• Nosotros jugamos – we play
• Ustedes / Ellos juegan – you all / they play

Spanish verbs are conjugated, and the ending -o in juego already tells you the subject is “I”, so you don’t need yo unless you want emphasis:

• Juego ajedrez. = I play chess.
• Yo juego ajedrez. = I (as opposed to someone else) play chess.

In Spanish, the verb ending usually makes the subject clear, so subject pronouns (yo, tú, él, etc.) are often omitted.

• Juego ajedrez. – the -o ending already implies yo.
• You add yo mainly for emphasis or contrast:
  • Yo juego ajedrez, pero mi hermano juega fútbol.
    I play chess, but my brother plays soccer.

So in neutral statements like this sentence, leaving off yo is the most natural choice.

Both exist; usage varies by region and style.

Traditionally (and especially in Spain), the standard pattern is:

• jugar a + [game/sport]
  
  • jugar al ajedrez (a + el = al)
  • jugar al fútbol

In much of Latin America, it’s very common and natural in speech to drop the a (and thus the article) and say:

• jugar ajedrez, jugar fútbol

So:

• Juego al ajedrez con mi abuelo. – fully correct and a bit more “textbook/neutral.”
• Juego ajedrez con mi abuelo. – also widespread and natural in Latin America.

Your sentence is fine for Latin American Spanish; if you want a very “safe” version understood everywhere as textbook-correct, use:

• Los sábados juego al ajedrez con mi abuelo…

Ajedrez (chess) behaves a bit like an uncountable noun when you mean the activity:

• After gustar, Spanish normally uses the definite article:
  • Me gusta el ajedrez. – I like chess.
• After jugar, in many varieties you can:
  • Use jugar a + el ajedrez → jugar al ajedrez, or
  • Treat ajedrez as a bare object: jugar ajedrez.

So:

• Juego al ajedrez. – literally I play the chess (game).
• Juego ajedrez. – I play chess. (game as an activity, no article)

You would not say un ajedrez for the game itself; un ajedrez would mean a chess set.

Spanish does not combine a definite article with a possessive before the same noun in modern standard usage:

• ✅ mi abuelo – my grandfather
• ❌ el mi abuelo – sounds archaic/poetic

You also usually include the possessive with close family members, unless the context already makes it obvious:

• con mi abuelo – with my grandfather
• con el abuelo – with Grandpa (used inside a family when everyone knows which abuelo you mean)
• con abuelo alone is unusual; it sounds like a nickname or vocative, not a normal object of con.

So con mi abuelo is the straightforward way to say with my grandfather.

Both relate to “board,” but they’re used differently:

• tablero

  • Specific “board” for games, instruments, dashboards, etc.
  • un tablero de ajedrez – a chessboard
  • un tablero de madera – a (structured) board / panel of wood

• tabla

  • A plank or flat piece of wood in general
  • A table/chart (as in data table)
  • Not the normal word for a chessboard

For chess, the natural word is tablero (de ajedrez), so:

• en un tablero viejo = on an old board (chessboard).
• en una tabla vieja would sound like “on an old plank of wood,” not specifically a chessboard.

Yes, adjective position often changes nuance in Spanish.

With many adjectives, including viejo:

• Noun + viejo usually describes an objective quality: old in age, worn-out.

  • un tablero viejo – an old/worn board (it’s old as an object)

• Viejo + noun can suggest long-standing / of long duration / dear, similar to “long-time” or “old (in the sense of ‘not new to me’)”.

  • un viejo amigo = a long-time friend, not necessarily an old person
  • un viejo tablero would more likely suggest a board you’ve had for a long time, with a more emotional or “long-standing companion” feel.

In your sentence, un tablero viejo neutrally states that the board is old (aged or shabby), which fits the most common interpretation.

Yes, Spanish word order is fairly flexible, especially for adverbial phrases (time, place, manner). All of these are grammatical, with slight differences in emphasis:

• Los sábados juego ajedrez con mi abuelo en un tablero viejo.
• Juego ajedrez con mi abuelo en un tablero viejo los sábados.
• Los sábados juego ajedrez en un tablero viejo con mi abuelo.
• Con mi abuelo juego ajedrez los sábados en un tablero viejo. (emphasis on with my grandfather)

The most neutral and typical for “On Saturdays I play chess with my grandfather on an old board” is probably the original, or:

• Los sábados juego (al) ajedrez con mi abuelo en un tablero viejo.

In Spanish, the simple present is the normal tense for habitual actions:

• Los sábados juego ajedrez…
  = On Saturdays I play chess… / I play chess on Saturdays.

Other options:

• Estoy jugando ajedrez con mi abuelo.
  = I am (right now) playing chess with my grandfather. (action in progress, no “on Saturdays” idea)
• Este sábado juego ajedrez con mi abuelo.
  = This Saturday I’m playing chess with my grandfather. (near future plan)
• Este sábado voy a jugar ajedrez con mi abuelo.
  = This Saturday I’m going to play chess with my grandfather. (future plan with ir a + infinitive)

For a general routine, present simple (juego) is exactly the right choice.