"I recently started playing the violin." Is saying that playing the violin is a new thing for you and that it is something you recently started.
"I just got into playing the violin." To be 'into something' means to have interest. Saying this example is saying you I just got interested in playing the violin. This is informal.
"I recently started playing the violin."（最近バイオリンを始めました）
↑バイオリンを弾くのは初めてで、最近始めたと伝えています。
"I just got into playing the violin."（最近バイオこリンに興味がでた）
「be into something」は「興味を持つ」という意味です。この文は：
"I just got interested in playing the violin."（最近バイオリンに興味がでた）
と同じ意味です。砕けた言い方です。
There are a few different ways you can talk about how you are learning to play the violin:
"I started playing the violin recently"
This is talking about how you just started learning not long ago. It is a very new thing that you started.
"I am learning to play the violin"
When you use this expression, you are saying you are in the process of learning. Maybe you are good at playing it or maybe not. But you are still learning and you are not an expert yet.
どのようにバイオリンを習っているか話すにはいくつかの方法があります。
"I started playing the violin recently"
最近、バイオリンを始めました。
これは、ちょっと前にバイオリンをはじめたことを伝えるフレーズです。最近新しく始めたということです。
"I am learning to play the violin"
バイオリンを習っています。
この表現を使うときは、習っている最中だということを言います。上手かもしれないし、そうでないかもしれませんが、まだ習っていて、まだエキスパートではないということです。
"I have started playing the violin."
This can mean you are learning how to play or that you just started practicing the violin for the day.
If you want to say you are learning to play the violin you can use the sentence below.
"I just started learning how to play the violin."
Learning - meaning you are taking lessons to gain the knowledge and skills to play the violin.
"I have started playing the violin."
（バイオリンを弾き始めました）
↑「バイオリンを習っている」もしくは「（その日の）バイオリンの練習を始めた」という意味になります。
バイオリンを習っていると言いたいなら、以下の文が使えます：
"I just started learning how to play the violin."
（バイオリンをちょうど習い始めました）
Learning - バイオリンを弾くための知識と技術を得ようとレッスンを受けている、という意味。
If you 'take up' something, it means that you start doing something - especially a hobby or pastime.
"John recently took up fishing and nowadays I hardly ever see him."
あなたが何かを「take up」というのは、あなたが何か、特に趣味や娯楽等を始めるということを意味します。
John recently took up fishing and nowadays I hardly ever see him.
（ジョンは最近釣りを始め、最近私はめったに彼を見ません。）
If you "take something up" it means that you start to practice something, normally a hobby. It's important to add the word "recently" so that your listener knows that this is something new. I hope that this helps. :)
Recently, I started learning how to play the violin
I recently started playing to violin
I've just started playing the violin
When you want to explain that you have just started playing the violin, then you may express this in the following way:
-Recently, I started learning how to play the violin
-I recently started playing to violin
-I've just started playing the violin
「最近バイオリンを始めました」は次のように言えます。
-Recently, I started learning how to play the violin（最近バイオリンを習い始めました）
-I recently started playing the violin（最近バイオリンを始めました）
-I've just started playing the violin（最近バイオリンを始めました）
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