Other corrections are useful. Here is just something to add:
有的服务免费，有的服务很便宜，有的服务很贵。For simplicity: 有的服务免费，有的很便宜，（还）有的很贵。
我有的时候网上买东西、飞机票，电子本书，音乐。我有时候在网上买东西：飞机票、电子书和音乐。
The differences between 网上、上网、网络购物：
上网 is a verb+noun structure; 网上 can be understood as a noun or a preposition. 网络购物 is a formal noun, sometimes can be used as a verb too.
So to correct this sentence, I would write:
我有的时候 上网买东西/上网购物/在网上买东西/在网上购物。我有的时候网上买东西 sounds strange, because 网上买东西 is a gerund and it needs 在 to make the sentence complete.
我有的时候进行网络购物 is grammatically correct, but sounds very strange too, because it is too formal.
网络购物 as a verb: 消费者(在)网络购物的时候，常常会不由自主地买很多东西。When online shopping, consumers can't help buying a lot of stuff.
但是我不喜欢买东西上网，我不相信。但是我不喜欢 上网买东西/网络购物/在网上买东西。I would use 我不信任它. Because 相信 is believe, but 信任 is trust.
因为上网有很多个骗子。网上有很多骗子。or 网上有很多骗局/网上有很多诈骗的现象。 "骗局/诈骗" is the direct translation of "cons", "骗子" is liar.
如果我不认识那个网站，我不用信用卡。 Grammatically correct but sounds strange, because 认识 is used for people. Here I would recommend 如果我不知道那个网站，我不会用信用卡。会 is to indicate the future tense in Mandarin, though you may not need a future tense in English.
不过我们上网可能买到很多很有意思的东西。
On internet there are a lot of services. Some are free, some are cheap and some are expensive. Sometimes I buy things on-line- plane tickets, eBooks, and music. But I don’t like online shopping. I don’t trust it because there are a lot of cons. I don’t use my credit card if I don’t know the site. However we can buy are a lot of interesting things on internet. There are more things on internet than in local shops.
online shopping
上网网上有很多服务。有的服务免费，有的服务很便宜，有的服务很贵。免费的服务网址常常有广告。我有的时候在网上买东西、飞机票，电子本书，音乐。但是我不喜欢在网上买东西上网，我不相信。因为上网网上有很多个骗子。如果我不认识那个网站我不用信用卡。
不过我们上网可能可以买很多很有意思的东西。上网网上比当地商店有东西的多。
On internet there are a lot of services. Some are free, some are cheap and some are expensive. Sometimes I buy things on-line- plane tickets, eBooks, and music. But I don’t like online shopping. I don’t trust it because there are a lot of cons. I don’t use my credit card if I don’t know the site. However we can buy are a lot of interesting things on internet. There are more things on internet than in local shops.
It is easy to get confused with "网上" and "上网".
Here are two sentences:
我上网看新闻。I went to the website and read some news.
我很喜欢上网。I like surfing the internet.
我在网上看新闻。I read news on line.
网上有很多商品。There are a lot of commodities on line.
I hope it helps.
online shopping
上网上有很多的服务。有的服务是免费的，有的服务很便宜，有的服务很贵。免费的服务网址常常有广告。我有的时候会在网上买东西，例如：飞机票，电子本书，音乐。但是我不喜欢在网上买东西上网，我不相信。因为上网上有很多个的骗子。如果我不认识那个网站，我就不用信用卡。
不过我们上网可能够买很多很有意思的东西。上网上比当地商店有的东西的多。
On internet there are a lot of services. Some are free, some are cheap and some are expensive. Sometimes I buy things on-line- plane tickets, eBooks, and music. But I don’t like online shopping. I don’t trust it because there are a lot of cons. I don’t use my credit card if I don’t know the site. However we can buy are a lot of interesting things on internet. There are more things on internet than in local shops.
online shopping
上网有很多服务。有的服务免费，有的服务很便宜，有的服务很贵。免费的服务网址常常有广告。我有的时候网上买东西、飞机票，电子本书，音乐。但是我不喜欢买东西上网，我不相信。因为上网有很多个骗子。如果我不认识那个网站我不用信用卡。
不过我们上网可能买很多很有意思的东西。上网比当地商店有东西的多。
On internet there are a lot of services. Some are free, some are cheap and some are expensive. Sometimes I buy things on-line- plane tickets, eBooks, and music. But I don’t like online shopping. I don’t trust it because there are a lot of cons. I don’t use my credit card if I don’t know the site. However we can buy are a lot of interesting things on internet. There are more things on internet than in local shops.
在网络所提供的诸多服务项目中，有免费的、有便宜的、有昂贵的。我有时候会购买线上机票、电子书、音源。但我并不喜欢在网上购物，这源于网络上大量的不诚信行为。我不会在不熟悉的网站使用信用卡。然而无论如何，我们可以在网上买到大量的物品，且其品种远多于下线的商店。
online shopping
上网上有很多服务。有的服务免费，有的服务很便宜，有的服务很贵。免费的服务网址常常有广告。我有的时候网上买东西、飞机票，电子本书，音乐。但是我不喜欢买东西上网，我不相信。因为上网有很多个骗子。如果我不认识那个网站我不用信用卡。
不过我们上网可能买很多很有意思的东西。上网比当地商店有东西的多。
On internet there are a lot of services. Some are free, some are cheap and some are expensive. Sometimes I buy things on-line- plane tickets, eBooks, and music. But I don’t like online shopping. I don’t trust it because there are a lot of cons. I don’t use my credit card if I don’t know the site. However we can buy are a lot of interesting things on internet. There are more things on internet than in local shops.
网上有很多服务。有的服务很便宜，有的服务很贵，还有的服务是免费的。但免费服务的网页上常常有广告。有时候我在网上买东西、飞机票、电子书、音乐等。但是我不太信号上网买东西，我不信任网络商店。因为网上有很多骗子。如果我不认识那个网站，我就不用信用卡。
不过我们在网上可以买到很多很有意思的东西。网上比实体店的东西多得多。
