There are several parts to a search engine. Broadly speaking, in a hopelessly general manner (folks, feel free to edit if you feel you can add better descriptions, links, etc):
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The crawler. This is the part that goes through the web, grabs the pages, and stores information about them into some central data store. In addition to the text itself, you will want things like the time you accessed it, etc. The crawler needs to be smart enough to know how often to hit certain domains, to obey the robots.txt convention, etc.
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The parser. This reads the data fetched by the crawler, parses it, saves whatever metadata it needs to, throws away junk, and possibly makes suggestions to the crawler on what to fetch next time around.
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The indexer. Reads the stuff the parser parsed, and creates inverted indexes into the terms found on the webpages. It can be as smart as you want it to be — apply NLP techniques to make indexes of concepts, cross-link things, throw in synonyms, etc.
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The ranking engine. Given a few thousand URLs matching “apple”, how do you decide which result is the best? Jut the index doesn’t give you that information. You need to analyze the text, the linking structure, and whatever other pieces you want to look at, and create some scores. This may be done completely on the fly (that’s really hard), or based on some pre-computed notions of “experts” (see PageRank, etc).
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The front end. Something needs to receive user queries, hit the central engine, and respond; this something needs to be smart about caching results, possibly mixing in results from other sources, etc. It has its own set of problems.
My advice — choose which of these interests you the most, download Lucene or Xapian or any other open source project out there, pull out the bit that does one of the above tasks, and try to replace it. Hopefully, with something better :-).
Some links that may prove useful:
“Agile web-crawler”, a paper from Estonia (in English)
Sphinx Search engine, an indexing and search api. Designed for large DBs, but modular and open-ended.
“Information Retrieval, a textbook about IR from Manning et al. Good overview of how the indexes are built, various issues that come up, as well as some discussion of crawling, etc. Free online version (for now)!