Saya ke dokter gigi.（I went to dental hospital.）

Hari ini, Saya ke dokter gigi.

There's no "Rumah sakit gigi" in Indonesian. So 鹿病院 is translated into 歯医者 which means "dokter gigi" :)

Saya memikirkan mau menahan sakit.

I don't really understand this sentence. Do you meant "痛いと思っていました”？If you are then the correct translation would be:

Saya berpikir akan sakit rasanya.

Tetapi, pengobatan hari ini tidak sakit.

Correct! 

Saya merasa baik - baik saja.

"Memikir" is not the right word to be used in this sentence.

これからもがんばってください！:)

Saya ke rumah sakit gigi.

Dentist = Dokter Gigi

so, the correct sentence is: Saya ke Dokter Gigi.

This is an informal sentence.

Hari ini, Saya ke rumah sakit gigi.

Hari ini, Saya ke Dokter Gigi.

Another suggestions:

1. Tadi siang, Saya pergi ke Dokter Gigi.

Since there is no past principle in Indonesian Language, I prefer to stated the time more spesific in sentence.

2. Tadi siang, Saya pergi mengunjungi Dokter Gigi.

This is formal sentence.

Saya memikirkan mau menahan sakit.

1. Saya pikir proses pengobatannya akan sakit.

This is formal sentence.

2. Saya pikir nantinya akan sakit.

This one is informal sentence.

There are still a lot of varieties to describe this in sentence, but I gave you two common options.

Tetapi, pengobatan hari ini tidak sakit.

This is already correct, but this way the sentence is more natural:

Tetapi, ternyata pengobatannya tidak sakit.

---nya refers to previous situation stated in the early sentence. In this case, Pengobatannya, ---nya refers to Pengobatan hari ini. This way, you can simplify the sentence and you don't have to mention several words twice.

Ternyata used if the sentence is describing an opposite fact or contrary from the previous sentence. 

Saya memikir baik saja.

Saya merasa baik-baik saja.