Yes, moving from the UK to Spain is realistic for a university lecturer and secondary science teacher, but you should plan around two hard realities: visa sponsorship and recognised qualifications. Because you are now non-EU citizens, you cannot just move and start working; an employer must sponsor you, and Spanish recognition processes can take longer than you expect.
Start with the visa. Spain has a work visa for highly qualified workers, sometimes called the HQS visa, and the EU Blue Card, both designed for employees with higher education and professional experience. In practice you need a job offer from a Spanish employer before applying. A chemistry PhD opens doors in industry, especially in pharmaceutical, chemical, environmental and energy companies, where English is more commonly used but Spanish is still a major advantage for everyday operations and integration. If you want to stay in academia, you would need to have your degrees formally recognised (homologación or the newer accreditation process), and university teaching in Spain is usually in Spanish or Catalan or Valencian or Galician, depending on the region. A permanent lectureship is hard to get even for locally qualified candidates, so expect temporary or research-only positions first.
Your wife's path is clearer. International schools in Spain regularly recruit UK-qualified teachers, and a secondary science teacher with QTS and a few years' experience is a strong candidate. The school usually handles the visa paperwork. Some schools also offer relocation packages. The tradeoff is that salaries are lower than UK teacher pay, and the best jobs are in Madrid, Barcelona, Valencia, Marbella and Mallorca. A school will also expect her to teach in English, so Spanish is not strictly required for day-to-day teaching, but it matters for life outside work, especially with bureaucracy.
Bringing a dog is manageable, but it is not a case of jumping on a plane. The UK is no longer in the EU, so you need an Animal Health Certificate, a rabies vaccination done at the correct time, and, for dogs, tapeworm treatment depending on the transport route. A vet who is registered to issue UK export documents can walk you through it. If flying, choose a pet-friendly airline and be careful with summer heat; many airlines ban pets in cargo on high temperature days.
Before moving, spend three to six months on this sequence. First, confirm your wife's eligibility for international school roles listed in Spain, and get her CV and references ready. Second, you should target chemistry industry jobs at large multinationals with Spanish sites, and contact academic groups in your field to test the water. Third, start structured Spanish lessons now, because even basic fluency transforms your options and your quality of life. Also build a realistic budget: Spanish salaries are lower, rent in coastal cities has risen sharply, and the tax system takes a noticeable share, though the pace of life, climate and access to the sea are usually the reasons people stay.