You can't buy a ticket only to the Roman Forum or only to the Colosseum. But you can visit several attractions at once, which is cost-efficient. There are three types of tickets:
— Integrated ticket Colosseum, Forum, Palatine – 16 euro, valid for 1 day.
— Integrated ticket Colosseum+Arena, Forum, Palatine – 18 euro, valid for 2 consecutive days.
— Combo Ticket Roman Forum-Imperial Forum – 18 euro.
You can buy ticket on the Internet or at the box office. For booking a ticket online you will have to pay 2 extra euro per person. Tourists under the age of 18 can attend the Roman Forum for free.
If you do not want to pay extra money, you can buy ticket at the box office. There are two of them: at the Colosseum and at the Roman Forum itself. As a rule, a huge queue is gathered at the amphitheater, and people, by inertia, also get up there and lose an hour (and sometimes more) of time. It is better to go a little further towards the Circus Maximus to the box office near the forum itself. The queues are not big there, and if you arrive in the morning, then you'll spend just a few minutes to get your ticket. The Roman Forum box offices are located on Via della Salara Vecchia, which starts from Via dei Fori Imperiali. You can get to the box office of the forum on foot from the Colosseo metro station.
To save money, it is better to buy a Roma Pass for 48 or 72 hours. Such card gives the right to visit 1 or 2 museum complexes (the Roman Forum among them), as well as freely use the city public transport. You can choose and buy a suitable card on the official website.