Learn Hardcore Malay: Sejak kami pindah ke bandar ini, kami lebih kerap makan di luar. - Since we moved to this city, we eat out more often.

Putting sejak... first sounds a bit more formal and gives more emphasis to the starting point/time.
Commas in Malay are flexible; you will often see the comma after the sejak clause when it comes first.

Why is it pindah and not berpindah here? Are both correct?

Both are possible, but pindah is more common in everyday speech.

pindah = to move (house, city, etc.), very common and neutral
berpindah = also to move, often sounds a bit more formal or bookish in this context

So you could also say:

Sejak kami berpindah ke bandar ini, kami lebih kerap makan di luar.

No big meaning change; it is mostly a style/register preference.

Why is it ke bandar ini and not di bandar ini?

Because the verb is pindah (move), which involves movement towards a place.

ke = to, towards (direction, destination)
di = in, at (location, no movement)

Compare:

pindah ke bandar ini = move to this city (destination)
tinggal di bandar ini = live in this city (location)

So after pindah, you normally use ke, not di, when you mention the place you move to.

Why is kami used, and not kita? Both mean we, right?

Yes, both mean we, but:

kami = we, but excluding the listener (you are not part of the group)
kita = we, including the listener (you are part of the group)

In Sejak kami pindah ke bandar ini..., the speaker is saying:

We (some group) moved to this city,
and the person being spoken to is not included in that we.

If the listener is part of the group that moved, you would say:

Sejak kita pindah ke bandar ini, kita lebih kerap makan di luar.

There is no past tense marker like telah. How do we know pindah is past?

Malay usually does not mark tense with verb endings. Time is shown by:

time words: sejak, semalam, tadi, nanti, etc.
context.

In this sentence:

sejak already tells you this event started in the past and continues to affect the present.
So pindah is understood as moved in English, even without telah.

You could add telah for a more formal, explicit past:

Sejak kami telah pindah ke bandar ini...

But in natural speech, telah is usually omitted here; sejak is enough.

What exactly does lebih kerap mean? Why not just kerap?

kerap = frequent(ly), often
lebih = more

Together:

lebih kerap = more often / more frequently

So the sentence is saying there has been a change: compared to before moving, they now eat out more often.

If you say only:

Kami kerap makan di luar. = We often eat out.

This describes a habit but does not clearly compare it with how it was before. Lebih kerap makes the comparison explicit.

Is kerap the only way to say often here? Could I use selalu or sering?

You have several options, all understandable:

kerap – common, slightly more formal or neutral.
sering – very common and neutral, many speakers use this a lot.
selalu – originally always, but in colloquial speech often used more loosely to mean often.

You might hear:

kami lebih kerap makan di luar
kami lebih sering makan di luar
kami selalu makan di luar (could mean always or very often, depending on context)

In this exact sentence, kerap and sering are the closest to more often.

Does makan di luar literally mean eating outside, or is it an idiom for eating out at restaurants?

Literally:

makan = eat
di luar = outside

But in everyday context, makan di luar usually means eat out (not at home), typically at:

restaurants
food courts
hawker stalls, etc.

If you wanted to emphasize physically outside in the open air, you would usually make that clear with more context, such as:

makan di luar rumah (eat outside the house)
makan di luar, di taman (eat outside, in the park)

In this sentence, the natural understanding is eat out (at food places).

Could I drop the second kami and just say Sejak kami pindah ke bandar ini, lebih kerap makan di luar?

Yes, in casual spoken Malay it is quite common to drop a repeated subject pronoun when it is clear from context:

Sejak kami pindah ke bandar ini, lebih kerap makan di luar.

This would be understood as:

(Kami) lebih kerap makan di luar.

However:

In careful writing or formal contexts, keeping kami is clearer and more standard.
For learners, it is safer to keep the subject until you are comfortable with when it can be dropped.

Does this sentence clearly imply a change in habit compared to before, or just describe the current habit?

It clearly implies a change.

The structure:

Sejak kami pindah ke bandar ini = Since we moved to this city (a starting point)
kami lebih kerap makan di luar = we eat out more often

Together they imply:

Before we moved, we ate out less.
After moving, our habit changed and now we eat out more frequently.

So the sentence points to a before vs. after contrast, not just a neutral description of the current habit.