Telling this from my own experience.

1. I work in a college, and WGU is recognized and even recommended by my superiors.
2. Many graduate schools accept and recognize WGU's Bachelor's and Master's. From my experience, the admissions personnel who respond to emails stating they don't recognize online schools are usually student workers. The only way for you to be sure is to actually apply and pay the fee. Then, it would go to a credentialing and evaluation officer who verifies your degree.
3. You can go directly from a Bachelor's to a Ph.D. However, you need to fulfill the school's requirements, such as minimum GPA, GMAT or GRE requirements, TOEFL/IELTS, etc. If you are applying from WGU (which doesn't provide GPA), you are almost certain to need to complete the GMAT or GRE first (unless waived).
4. Other requirements you need to fulfill include completing certain courses. For example, if the Master's program requires programming or higher-level math knowledge, they might reject you and ask you to take some sort of class to satisfy this requirement at their school.
5. When comparing online versus Brick-and-Mortar (B&M) Master's/Doctorate programs, B&M schools are almost certain to waive your tuition and provide stipends, but you need to teach or do research for them. However, you may still need to take the GMAT/GRE, though some schools do waive it. Online schools usually never have any scholarships or stipends.

Additional suggestions:

1. Research Experience: Prior research experience, whether through internships, independent projects, or collaborations with faculty, can significantly strengthen your application. Highlight any research you've conducted, including publications, presentations, or posters.
2. Statement of Purpose: Craft a compelling and well-written statement of purpose that outlines your academic interests, career goals, and why you're interested in pursuing a Ph.D. at that particular institution. Be specific about how your research interests align with the faculty's expertise at the program.
3. Letters of Recommendation: Choose recommenders who know you well and can speak to your academic abilities, research potential, and personal qualities. It's often beneficial to have letters from faculty members who have supervised your research or coursework.
4. Fit with Program: Demonstrate a strong fit with the program by researching the faculty's research interests, ongoing projects, and recent publications. Tailor your application to showcase how your academic and research background align with the program's strengths.
5. Graduate Record Examination (GRE): While some programs may not require the GRE, others still do. Prepare thoroughly for the exam if it's required, and aim for scores that are competitive with those of admitted students in your desired program.
6. Interview Preparation: If interviews are part of the application process, prepare by reviewing your research experience, academic background, and reasons for pursuing a Ph.D. Be ready to discuss your research interests and how they align with the program.
7. Professional Development: Highlight any relevant professional development experiences, such as teaching assistantships, leadership roles, or participation in academic conferences or workshops.
8. Writing Sample: If required, submit a writing sample that demonstrates your ability to conduct scholarly research and communicate your ideas effectively. Choose a sample that showcases your analytical and writing skills in your field of study.
9. Funding Opportunities: Research funding opportunities available within the program, such as fellowships, assistantships, or grants. Highlight any relevant funding you've received or your willingness to seek external funding to support your Ph.D. studies.
10. Follow Up: After submitting your application, consider following up with the admissions office to ensure all materials have been received and to express continued interest in the program.