March Comes in Like a Lion...

There are were also heavy snowfall areas in the northern Japan.

This kind of weather always reminds me of an old English saying, "March comes in like a lion and goes out like a lamb." I like this expression because it is admirably put.well stated.

After a long cold winter, many of us are waiting eagerly for a spring, but it usually doesn't come right away.

Many people were strolling around a shopping mall, a park or a sight seeing place and enjoyed lots of sunshine and a mild weather.

We have an expression, "春の嵐"（haruno arashi)" that means "spring storms".

You put the quotes around exactly what you're emphasizing, so the parenthetical is left outside.

If you put the last quote before the period, that's British-style punctuation.

Even in the middle of March, we had some very cold days that felt just like winter.

I would prefer saying that they feel like but were not just like winter.

Some places in northern Japan are still getting a lot of snow.

I made it more casual. If you want a very formal sentence, the correct way would be: "There are also areas in Northern Japan with heavy snowfall."

In either case, put "in Northern Japan" closer to "areas/places" to make it clearer.

Every (single) day, they have to remove (accumulated) snow from their roofs (in order) to protect their houses.

Every "single" day is a little dramatic for this.
"Remove X from Y."
You can take out the blue words to make it more casual.

This kind of weather always reminds me of an old English old saying, "March comes in like a lion and goes out like a lamb." I like this expression because it is so (neat and accurate / cute but true).

Old should come before English. (Word order.)

I'm not sure what you mean by "admirably put." So, I came up with two possible alternatives.

After a long cold winter, many of us are waiting eagerly for a spring, but it usually doesn't come right away.

No need for "a."

Today, March 20th, is the Vernal Equinox Day, a national holiday.

Lots of people were out in the malls and parks, enjoying the sunshine and a mild weather.

"Lots of" is more casual.
You use plurals for general things, like "stores" or "hospitals."
I couldn't think of a natural way to refer to "sightseeing spots," so I took it out.

My family and I also went to see the colorful plum blossoms in a nearby garden.

I think it sounds more natural to mention where the garden is, rather than what kind of garden it is. Even if they only have plum trees, it seems redundant to say a "plum garden." If you're going to see plum blossoms, then they must have plum trees. :}

I love these signs of spring and in a few weeks, we are going to see gorgeous cherry blossoms everywhere.

"They are truly" is quite dramatic. ^_^;

Now I think Spring is just around the corner.

You said there were already signs of spring, so you shouldn't say "I think."

There was also heavy snowfall areas in the northern parts of Japan.

Even in the middle of March we had some very cold days that felt like winter.

There were also heavy snowfall areas in the northern Japan.

Every single day, they have to remove accumulated snow from their roofs in order to protect their houses.

This kind of weather always reminds me of an old English saying, "March comes in like a lion and goes out like a lamb." I like this expression because it is well put.

After a long cold winter, many of us are waiting eagerly for spring, but it usually doesn't come right away.

Today, March 20th, is Vernal Equinox Day, a national holiday.

Many people were strolling around a shopping mall, a park or a sight seeing place and enjoyed lots of sunshine and mild weather.

My family and I also went to see the colorful plum blossoms in a nearby plum garden.