Around the world, there are rocks scattered all over the place. If you click on them an dig, sometimes you will obtain crystals and metals. Another way to obtain crystals and metals is by asking your fellow scientists for them.
Does the invention ask for crystal and metals, or specifically elements? If asking for the former, all you need to do is what I explained above. If it's the latter, you need to do one of two things. One option is to send those crystals/metals to the geo counsel, which you can do by clicking on them on your home lot. In a few days you will receive elements in the mail. Another option is to analyze those crystals/metals using the chemical analyzer. Sometimes while doing so you will receive a new element. Less reliable but more time effective than the mailing method.