The difference between đi, đi đến and đến

In general, đi means "to go" and đến means "to arrive". But there are some more nuances. First, Vietnamese doesn't distinguish movement toward the speaker ("come here") and away from the speaker ("go there"). In either case, đi is used. That can be quite confusing for speakers of Western languages. For example, "I'm coming!" is "Tôi đang đi!".

The word đi has the following applications:

1. It denotes movement. 

There is no object after đi in this case.
Đừng đi nhanh quá! (Don't go so fast!)
Thôi, trễ rồi. Em đi. (It's late already. I'm leaving.)

2. It is equivalent to the expression "to go + [verb]-ing" in English (go swimming, go hiking, go dining).

A verb follows after the đi.
Tối qua anh đi chơi với bạn bè. (Last night I went out to have fun with friends.)

Cô ấy thường đi leo núi. (She often goes hiking).

3. In certain expressions in combination with a noun, like "đi chợ", "đi bác sĩ" or "đi bộ đội".

Here, đi shouldn't be understood as "go", but rather as part of the expression denoting an action. Like, "đi chợ" is "to buy stuff at the market" and "đi bác sĩ" is "to see a doctor". Expressions like "đi McDonald's" are also often used, although this might not be considered correct by purists. Here, again, this can be understood as "eat at McDonald's" rather than "go to McDonalds".

4. It can be used for any geographical location (country, continent, city, mountain, sea, ...).
Họ không có tiền đi nước ngoài. (They don't have money to go abroad.)
Đi biển đi! (Let's go to the beach!)

The expression đi đến, on the other hand, is used to denote movement to a specific location (other than geographical locations as described above). Here are two sentences contrasting đi and đi đến.
Học sinh này đang đi đến trường gần nhà. (This student is on their way to school near their home.)
Học sinh này đang đi trường gần nhà. (This student is attending a school near their home.)

Finally, đến (or tới) has two use cases.

1. It means "to arrive".
Tôi đến rồi! (I have arrived already!)
Mùa xuân đến rồi đó. (Spring has arrived.)

2. It is short for đi đến.

So, instead of...
Học sinh này đang đi đến trường gần nhà.
...we can say...
Học sinh này đang đến trường gần nhà.