How I can choose an engagement ring without burning a hole in my pocket?.

There are certain guidelines one can follow. Many brides-to-be will have to decide what is in it that they want when buying a ring engagement. You will need to consider the cost, which is the foremost rule and a basic one to follow.
Avoid the notion that big and quantity is all you have to it but paying more than what you can afford is certainly something you need to consider and be on-guard. It could set you into a trap that later might leads to financial difficulties which is not going to do any good to the relationship. Keep it simple. It would be better off making ring purchases within your means rather than beyond your means.
Why pay above your means for an engagement ring that could only put you in debt. It is wiser to work around your own budget and not somebody else’s budget. Try to work along this line and you won’t go wrong. Have an open and honest discussion with your fiance, then proceed to pay for what you both are comfortable with.
If you go with quality over quantity, then it is unlikely you will go wrong. The rule is pay only what you can comfortably afford. For example, you don’t need to spend three months’ salary for the ring if you cannot afford it. Do not be overtaken by overzealous talk from marketing or salespeople into paying more than what you can afford nor be cowed into feeling less of a person if you do not go for more.
Just beware of the marketing ploy by big companies and know your real needs. No need to put yourself into unnecessary stresses but instead keep it as simple as you can, following the above rather straightforward principles in going about your engagement ring buying. What you can afford at the moment, is what you should be focusing at.
Upgrade later where chance permits later on in life. Any extra you may have now could be utilized for some other useful household matters such as upgrading of home decor. This is to keep a balance, so you can also focus on other equally important matters which may also require financial resources, just in case, if you have to, in order to meet any shortfalls later. In this way, you can work out your ring engagement buying in a properly manageable position.
Endless choices of fine selection is very much available for those who have a fancy for varied specialty designs and a budget concern to look out for; and you can find all these among the ‘White Gold Engagement Ring’ category. For example, 14k white gold ring is already a hit among wearers of different age group, especially in the western world. Perhaps, this can be one option you can look into. Hope this little guideline helps.