が、とても

but very

→ が
phrase Rare #30,739
but very が connects contrasting ideas and とても means “very”; together, they introduce a strong contrasting description.
JA: この店は小さいが、とても人気がある。
EN: This shop is small, but very popular.

幾何学について私は何も知らないが、とても退屈な学科らしい

I know nothing about geometry, but it sounds like a very boring subject.

彼は若いが、とても礼儀正しい

Although he is young, he is very polite.

彼女をデートに誘いたいんだが、とてもその勇気がない

I want to ask her out, but I just don't have the courage.

彼の奥さんには駅であったのだが、とても魅力的な人だ

I met his wife at the station, and she’s a very charming person.

コンサートは短いものだったが、とてもよかった

The concert was short, but it was very good.

日本製の時計を持っていますが、とても時間が正確です

I have a Japanese-made watch, and it keeps excellent time.