The companies sitting with fleets of vehicles to rent out have set packages and want you to fit into one of them. By default, if the booking is imprecise, you'll find yourself arbitrarily dumped into their "standard" vehicles on the "usual" terms. Perhaps this will work well for you. But when it proves a bad guess, you need to unpick the agreement when you arrive at the collection point without causing a major war. Remember you need the company to give you a vehicle at a good price so upsetting the local representatives before you start is not the best way to win. Indeed, you should have done all the heavy lifting by email before you set off for the collection point.
Start with the question of how long you want the vehicle. If this is literally one or two days, you will nearly always do better with one of the national or international companies. The large companies always carry a big fleet of vehicles of different sizes so can more easily fit you in for short periods. When you approach a "local" company, it will want you to take the vehicle for a minimum period, usually a week. If you insist on a shorter period of hire from a small company, you will be denied any possibility of a discount. Companies that rent out on predictable cycles can be left with a vehicle sitting in the lot waiting for the start of the next period of hire to start.
Now ask yourself how far you're proposing to drive. If this is going to be within the immediate area, going local will be safe. Should you breakdown or hit problems, people can come and rescue you. But if you're planning to take off into the boondocks, a national company is going to give you more protection because, hopefully, you will always be within range of a branch office or the national company can tap a local garage to give roadside assistance. Alternatively, ensure you have membership of the national roadside rescue services in the countries you propose to visit. You always need to be just one cell phone call away from help.
To get the best deals, always ask about discounts. It's a rule of life. If you don't ask, you never get a discount. If this is going to be a regular event, ask about a loyalty program. Most are free but, some times, you pay a small amount to join. Once in, you get a discount on all the bookings you make. The same applies if you're already a member of frequent flier programs or your credit card provider has a local promotion. In the other direction, always ask about any terms entitling the car rental companies to impose charges or fees. For example, what are the no-show rules? Some companies charge you only if you fail to collect at all. Others charge you if you're late. So, if you're stuck in traffic on your way to the collection point, are you calm or is there frustration because a fee will be imposed unless you can get there in the next five minutes? Car rental companies like the chance to add to your bill. Know what you're getting into before you sign up.