A vampire don't always travels alone. Keeping some help around can prevent easily getting a stake through the heart while they rest. Vampires are versatile monsters, and can be found in many different situations. Here I outline a group that a vampire might use as they travel, or flee for their life after a setback.
4 Bandits using short swords, and crossbows
1 Carriage with 2 horses
1 Vampire Spawn
* Find stats for the Vampire Spawn (page 354) and Bandits (page 396) here.
The vampire will typically either use it's minions that it travels with as a source of blood to stay alive, or use them to aid in staging attacks. The chosen strategy depends on the disposition of the vampire in question, with those with poorer self control and those enjoying the hunt praying on innocence instead of their minions. A vampire that can subsist on its minions will typically develop schemes to hide its tracks and goals that reach beyond their basic needs.
Using the carriage and its minions, the vampire will either set up a base and use the carriage as a quick escape if things get too suspicious, or travel over a large area to spread out the attacks. The minions watch over their master's hide out, spacing out their rest to ensure there is always at least one lookout. The presence of minions means that even during the daytime, the vampire can be moved.
Due to a vampire's forbiddance, and the legends that people at large are often aware of, it's not uncommon for villagers to stay indoors at night once a vampire is suspected. For a vampire who enjoys the hunt, they need some creativeness to overcome the issue. The most common of which is to change hunting grounds, to have spies operating in the area who gain the trust of locals and invite the vampire in, having their minions use force to break into a house containing potential victims and then inviting the vampire in, forcing occupants out of a building and attacking them when they leave, or for the vampire to invite people over to their home for parties in the attempt to elicit an invitation (it's not good form to kill people in your own house unless you can get away without being suspected).
Different vampires will have different numbers of minions. In this group I've set up a small group that transport the vampire. However, it can have more minions planted in the village or city. It can also have the addition of an honour guard if need be. It could also be moving between areas of large influence. Depending on its goal and situation, the behaviour can change greatly.
A vampire could spend a great deal of time working its way into the good books of local nobles and influential people. They could even play the part of commoner to ensure no-one with power looks at them twice. This means that finding out who the vampire is could be just the beginning. The players may need to deal with innocent guards doing their job, characters who simply don't believe their wild accusations, and many different forms of backup the vampire can raise.
Things Not Going as Planned
A vampire spawn knows that it can die, and will try to preserve itself if in distress. However, it also knows that it has tremendous regeneration abilities and may come back minutes later, ready for more while its targets are low on resources. Regardless, it will try its best to preserve itself and the location of its resting place, fearing the possibility of being staked.