The tires will not 'pop' because you are a big person, if that is your concern. However, you do need to be proactive about keeping the tires inflated nice and firm - probably over 100psi for 25mm tires. Riding with underinflated tires (the definition of 'underinflated' depends on the rider weight, road conditions, riding style, etc) is very likely to result in 'pinch flats' (tire and tube getting pinched between the ground and the rim, damaging the inner tube) or damaging the rim.
If you can fit 32mm tires then do so - you can probably run them around ~80-90psi, they will roll lightning fast and give good comfort, traction, and rim protection.
The cost of a decent tire starts around ~$30 each. More expensive tires can either be faster rolling and lighter, or more resistant to damage and punctures and a bit slower. If you were strapped for cash there are also perfectly good tires for $20 or less each. My absolute favourite tires are Panaracer Pasela - available in lightweight and supple fast rolling, and slightly less fast puncture resistant versions. The fast ones are usually sold for about $25 each and they come in many many sizes!
The great thing about tires is that they wear out if you ride a lot, so you can reward yourself for riding a lot by buying new tires!