If you are new to the research process, start with the FamilySearch Wiki's article on the Research Process. Start a notebook and research log to write down what you already know and what you want to learn about, so you can keep track of what you've done and see your progress.
When you are trying to trace your family members who immigrated to the United States, it helps to do as much research as you can in the United States before you try to jump back to their original country. It sounds silly, but you can make better progress by going forwards before you go back.
Gather all the information you have about your mother's life in the United States and look at all of it as a group. Establish a timeline for her and think of specific questions you can ask (and answer) to fill in her timeline. Checklists can help you think of records that you might have missed. Making a Genealogy Source Checklist like the one demonstrated in this video by Crista Cowan will help you stay organized.
You could ask:
- Did your mother become a naturalized citizen of the United States, and if so, when?
- When did she arrive in the United States?
- Did she make more than one trip to the US before she came to stay? Do you have all the arrival lists?
- If you have her arrival lists, what associated persons (relatives in their old country, and 'going to meet' people in the United States) are listed in her entries? Read the entire entry and see what clues you can find.
- If you have her arrival list, did she travel as part of a group? Did she buy her own ticket, or was it purchased by someone else?
- Did your mom have siblings? Can you find records in the United States for them? Sometimes we can't find information in the records about our direct ancestors, but we can fill in gaps from the records of their brothers and sisters.
Keep track of all the names you find in your mom's records -- you may need to learn more about all of those people and study the entire group to find the right place in Peru. See the Wiki article: Tracing Women Using Cluster Genealogy.
Use tools like the United States Record Finder to focus your search on record types that have the information you're looking for.
Once you have more information and are ready to search in Peru, before you leap into searching records on FamilySearch, Ancestry, or elsewhere, consult the FamilySearch Wiki's articles on Peru Genealogy. The main page has links to Research Strategies, Getting Started in research in Peru, to Peru Online records, and to video classes in FamilySearch's learning center.
The Peru Record Finder can help you find which records will help you answer your questions.
A few other basic principles to keep in mind when dealing with historical records:
- Most records are kept because a law was passed saying that the records should be kept. Sometimes knowing the law can help you figure out whether your people should be in the record group or not.
- When looking at record groups in an archive, keep in mind the "three-legged stool" -- if you know a name, a place, and a time frame, you have a solid place to stand. If you know only two of those things, your starting point for the search is more uncertain.
- The FamilySearch Wiki has an article about each collection of records which explains what kind of information you might find in the records.
- Be aware that not every bit of the information in a record is indexed. Sometimes it is better to start with a wide search and then to narrow it down.
The other thing you need to keep in mind when doing long-distance research is that not everything is online. However, new records are coming online all the time, so you have to keep going back to sites to look for what has been added.
For FamilySearch, you can:
- Look for posts about Peru in the FamilySearch Newsroom
- Review the Collection List (Browse All Published Collections) and note the Last Updated dates. You can filter by location on the left.
- If you use the Peru Research Page which was mentioned in the previous answer, be aware that not all of the collections may be listed there. When collections are being indexed, they may fall off the list because they aren't completely indexed, and they aren't completely browse-only. Use the FamilySearch Catalog and do a place search to be sure you've looked at everything, and be sure to check all jurisdictions to make sure you haven't missed anything.
- Go to the Card Catalog, which can be reached from the drop-down menu under Search on the Main Page. You can sort the list by Date Added or Date Updated, and filter by location.
The same principles can be used when you move to doing research in Japan or Spain. Start with a person's death (or probate) where the records have the most detailed portrait of their life, and work backwards along their timeline. Work from the known to the unknown, as much as you can, rather than trying to make huge leaps.