How has my company’s SEO strategy and tactics changed since Penguin and Panda?   Quite a bit, but many of the basics are still the same.  Below is a very brief and non-detailed before and after picture of our SEO.   There are a lot more things we look at and work on, but below should give you the gist of the changes. 
Penguin and Panda are two major classes of Google algorithm changes that have literally set the SEO world on its ear.   What worked several years ago does not work today and can even get you banned. Penguin mainly deals with inbound links and on-site optimization.   Spamming or low quality links or excessive optimization will hurt your site.    Panda is more related to content. Duplicating content or scraping the internet to create content will hurt your site. Not having enough content can be a determent.
Below are the basic steps we used when doing SEO on a website. I tried to keep it short, but I kept on expanding.
Pre Penguin/Panda
Site Assessment:  Analyze current inbound linking situation, site structure issues, page rank distribution, urls. Review current ranking profile and history. Assess how strong your inbound links are compared to competition.
Keyword Research: Find keywords that have good search volume, with minimal competition and high relevance.  Work with clients to pick keywords.
Optimize Content: Add more content, optimize existing content: title tags, Meta descriptions, alt tags internal linking and make sure that text links have keywords. Add html and xml site maps and set up with Google and Bing webmaster tools.
Link Building: Directory submissions, article writing, press releases, forum posts, buy links from high profile bloggers, try to get links from associations, distributors and a whole host of other link building schemes has been used effectively.
Blogging or other site content generation was recommended, but not heavily pushed.
Post Penguin/Panda
Site assessment – similar to before accept may need to dig deeper to determine any penalties from Penguin or Panda. Looking at mobile friendliness, download speed, review internal linking, and ensure that site is not over optimized.   Use tools that can assess historical ranking trends.    Review Social Media, citations, customer reviews and online reputation. Review consistancy of online profile across web properties.
Keyword Research: Analyze search behavior and determine gaps between website content and search terms that are being used.   Tools are not as precise as they were years ago, but we are taking a broader look at what search trends are rather than focus on precise search terms.
Optimize Content: Title tags are still very important, focus more on content than the technical coding,   need to avoid over optimizing,   reduce emphasis on text links, don’t worry about page rank, avoid internal no follow tags.   Probably still need to pay attention to the details as before, but they don’t have as much impact. Set up Google analytics, Google and Bing webmaster tools and submit xml site maps.   Addressing Mobile friendly issues and possibly look at ssl for entire site.
Link Building: This area is one of the biggest changes, don’t do most of the techniques listed above, getting some links from highly relevant and authoritative  directories maybe okay, but they should be few and far between.  We try to focus on attaining “natural links”. 
Branding: This is a broad category of effort, but it includes social media, customer reviews and other citations. None of the effort really builds links, but you are trying to connect directly with your customers by engaging in relevant conversation with them, and promote your brand to your market. Social media alone is a huge undertaking, but probably is needed to make in the post penguin world. This is a big change from pre Penguin and Panda.
Content Generation: In short, upload, unique, informative, highly relevant and interesting content on your site and promote that content through your social media channels.   The more posts the better, but make sure the content is extremely good.   This strategy is probably the safest method of link building. If enough people see your highly informative content, the better chance of getting some links.   Even without the potential for link building, content generation will increase your traffic, probably improve your rankings and supplement your branding efforts. We tell our clients that content generation needs to be a critical consideration for internet marketing.
In short, SEO efforts today are much broader in scope, than years ago, Google is looking at a lot more factors outside of the website and it appears that they will continue that trend. Website usability and user experience is critical and should constantly be analyzed.