As per subject, anyone can tell me how to check that my site is getting index/rank by search engine(e.g. google, yahoo)?
To see if you are indexed just search for your domain name. If you appear then you are indexed.
To check to see how you rank requires doing searches for terms you expect to be ranked for and then see if you appear. Of course, you may not rank well and thus not find your pages being listed. That just means your pages rank poorly. See this question for tools to help automate this process.
Google Webmaster Tools allow you to see specifically what is being indexed on your site, when, and what the results are.
If you talk only about google which is primarily the main source of web search traffic for most of the web, then google provide a tool known as Google Webmaster that gives u following data points:
1) Impressions for your site<br/> 2) Clicks<br/> 3) Average position in search<br/> 4) You can get brand vs non-brand traffic<br/> 5) You get number of pages ranked, crawler frequency, errors etc<br/><br/> + a lot of unending features
But if you want to analyze traffic from other sources as well, then there are again many options to do so. I will be listing few of those:
1) If only traffic is required, then google analytics tool provide you the referral of search (i.e. search is from which search provider). GA has free version + a better premium version if you can afford.
2) If you want to go for in-depth analysis, then tools like RankWatcher, SimilarWeb, GetStats might help you analyze search traffic + your standing w.r.t. competition. (but these tools are paid)
So you might look for a tool based on your requirement.