Сегодня мы едем в центр на трамвае, а по выходным часто ездим туда на метро.

Both come from the verb pair ехать / ездить (to go by vehicle).

• ехать → едем = one-time, right now / today, movement in one direction.

  • Сегодня мы едем… = Today we are going (this one trip)…

• ездить → ездим = habitual / repeated action, from time to time, usually.

  • …по выходным часто ездим… = on weekends we (often) go (regularly)…

So the sentence contrasts:

• едем for today’s specific trip, and
• ездим for the general, repeated weekend habit.

The preposition в can take either:

• Accusative (direction: to, into, toward), or
• Prepositional (location: in, at).

Here we have:

• в центр = to the center / downtown (direction → Accusative: центр → в центр)
• в центре = in the center / downtown (location → Prepositional: центр → в центре)

In this sentence we talk about going to the center, so в центр is required.

With means of transport, Russian normally uses на + [Instrumental case]:

• на трамвае (Instrumental of трамвай)
• на автобусе, на машине, на поезде, etc.

Метро is indeclinable (its form doesn’t change), so на метро looks like Nominative but is understood as Instrumental.

Semantically, на + Instrumental here means by / on (a means of transport):

• ехать на трамвае = to go by tram
• ехать на метро = to go by metro / subway

По выходным literally is along / over weekends, and idiomatically means on weekends / at weekends (regularly).

Grammar:

• выходной день → plural выходные (дни) = weekends / days off
• по выходным uses Dative plural: выходные → выходным

Typical pattern:

• по понедельникам – on Mondays
• по вечерам – in the evenings
• по праздникам – on holidays

So по + Dative plural expresses regular repetition on those days.

Both exist, but they’re not identical:

• по выходным — focuses on regularity/habit: on weekends (generally, as a rule).
• на выходных — usually about a specific weekend or weekend time frame: this/that weekend, on the weekend.

In this sentence, we’re describing a habit:

• On weekends we often go there by metro → по выходным часто ездим… is the natural choice.

Туда means there (direction) and refers back to в центр.

• First clause: в центр specifies where we are going today.
• Second clause: we don’t repeat в центр, so we replace it with туда = to that place (there).

So:

• Сегодня мы едем в центр…
• …а по выходным часто ездим туда… = …but on weekends we often go there… (to the same place just mentioned)

You could repeat в центр, but туда avoids repetition and sounds more natural.

Both а and и can be translated as and, but:

• и = simple addition, and also
• а = contrast, comparison, whereas / but / on the other hand

In the sentence, there is a contrast:

• Today (one-time action) we go by tram.
• On weekends (habitual) we go by metro.

So а correctly marks this contrast:

• …на трамвае, а по выходным… на метро.
  = …by tram, whereas on weekends… by metro.

Russian has two sets of motion verbs:

1. On foot:

   • идти (one direction, one trip now)
   • ходить (habitual / back-and-forth / multi-directional)

2. By vehicle/transport:

   • ехать (one direction, one trip now)
   • ездить (habitual / multi-directional)

In this sentence, we are clearly using transport (tram, metro), so we must choose from ехать / ездить, not идти / ходить.

• Сегодня мы едем… (by tram, one trip)
• …часто ездим… (by metro, regularly)

Each preposition requires a specific case:

1. в центр

   • в
     • Accusative = direction / destination
   • noun: центр → в центр (Accusative singular)

2. на трамвае

   • with means of transport: на
     • Instrumental
   • noun: трамвай → трамваем (Instrumental singular)
   • after на, Instrumental is realized as на трамвае (fixed phrase form)

3. по выходным

   • по
     • Dative (for repeated times/days)
   • noun: выходные → выходным (Dative plural)

Different roles → different prepositions → different cases.

Yes, Russian word order is flexible, and these variations are possible:

• Мы сегодня едем в центр на трамвае… (neutral, slightly more emphasis on мы)
• Сегодня мы едем… (neutral, typical narrative start: Today we…)

Second clause:

• По выходным мы часто ездим туда на метро.
• Мы по выходным часто ездим туда на метро.
• Часто по выходным мы ездим туда на метро. (emphasizes often)

The original order is natural and neutral; other orders add slight emphasis but remain correct.

Часто (often) is an adverb of frequency and is fairly mobile, but it normally appears near the verb or time expression.

Possible positions:

• По выходным часто ездим туда на метро. (neutral; often linked to weekends)
• Часто по выходным ездим туда на метро. (often slightly emphasized)
• По выходным ездим туда на метро часто. (less natural; sounds “tacked on”)

The original по выходным часто ездим is a very natural placement.