2000 Rover 25 not starting

Had a 2 incidents where my car wouldn't start, completely dead. It had to be jump started to get it going and the battery was put on charge. It charged very quickly which puzzled my retired mechanic dad but he suggested I prob needed a new battery as it may have a dodgy cell and it hadn't been replaced since the car was new. I got a new one fitted and the garage said it was on its way out anyway. Its been fine for 2 weeks and today it has done it again. Luckily outside my house and neighbour helped me push onto the drive, then when I tried it again it started, did another short journey and now completely dead again on the drive. All the lights come on, radio comes on b but engine does not turn over at all. Could it be the mobiliser playing up or the starter motor? Getting a jump start in the morning and taking it to my garage to check it out but would be grateful for any info from anyone, and if its starter motor is it an expensive job? threatening to get a new car but she had been a trusty companion for 10 years with low mileage so am reluctant to let her go!!