Big 6 matches (Arsenal, Chelsea, City, Liverpool, Spurs, United) sell out in minutes. Your best bet: membership (£25-50/year) gives you access to the general sale. Hospitality packages start at £200-500. For tourists: Crystal Palace, Brentford, and Brighton are easier to get tickets for and have great atmospheres.
Ticket: £30-60 (general) to £100+ (premium). Pint at the ground: £6-7. Pie: £4-5. Programme: £3.50. Total realistic budget for a day out: £80-120 per person. London grounds are 20-30% more expensive than northern venues.
Anfield (Liverpool): You'll Never Walk Alone before kickoff is a religious experience. St James' Park (Newcastle): 52,000 people who care more about football than anything. Selhurst Park (Crystal Palace): The loudest per-capita ground in the PL.
Arrive 45 minutes early for the atmosphere. Don't wear the wrong colors in the wrong stand. Standing is normal even in "seated" sections. The pubs around the ground are part of the experience — get there 2 hours early. Cash is rare; everything is contactless.