What I've done in my Hoard of the Dragon Queen Campaign is to have "the local clerics/priests" resurrect dead characters in exchange for the assistance the party is providing the town. So, if your characters are helping the city (I haven't read the starter set, avoiding spoilers), then local authorities could provide healing in exchange. This means, however, that the character has to sit out the next mission (so, they'll have to roll a temp or something--temp could be a generic NPC, like a soldier from the local militia, sent to help the party; or they could roll another character, though it has to be a different race and class than the one that died).
If they actually have the resources to pay for the spell (as per the Starter Set Guidelines), then the character doesn't have to sit out. I think it's a fair 'penalty' without trivializing death.
More specifically, Neverwinter has traditionally been one of the largest (if not the largest) cities in the Sword Coast--undoubtedly such a service is available. The cost would be at least 500 gp, given the material components required to cast Raise Dead.
Then again, the city was presumably destroyed in the 4E lore; without having read the starter set info (I don't want to spoil it), I'm not sure about the current state of the city. This wouldn't be a random NPC; however, it need not be a "high-level cleric--" a L9 Cleric can cast Raise Dead.
About hiring spellcasters, from the PHB, p. 159:
Hiring someone to cast a relatively common spell of 1st or 2nd level, such as cure wounds or identify, is easy enough in a city or town, and might cost 10 to 50 gold pieces (plus the cost of any expensive material components). Finding someone able and willing to cast a higher-level spell might involve traveling to a large city, perhaps one with a university or prominent temple. Once found, the spellcaster might ask for a service instead of payment— the kind of service that only adventurers can provide, such as retrieving a rare item from a dangerous locale or traversing a monster- infested wilderness to deliver something important to a distant settlement.
It is a problem that Lost Mines of Phandelver is designed for 3-5 player characters (PCs), plus a DM. I'm playing Phandelver at the moment with a group. One session only two players turned up, and I discovered how deadly this could be with no modification. With one PC and no modification, you would likely not last past the first encounter. As far as I can see it, you have options:
1) Tone down Lost Mines of Phandelver
By 'tone down' I mean reduce the level of the encounters. This will be easier near the beginning (where you could just reduce the number of goblins), and more difficult as the game progresses.
I actually did this with the session in question - they were supposed to encounter 6 hobgoblins (rolling on the random encounter table at that point in the game), but I cut it down to two.
With bigger monsters it is more difficult, though you could 'wound' them, reducing their HP (hit points) and possibly some of their special abilites, and XP (experience points) commesurately, but that can get complicated and possibly lack verisimilitude, so there is always option 2).
I didn't do this to one monster (a grick) and both characters ended the session unconscious. Oops.
2) Play Phandelver, but have your girlfriend control two or more characters, and/or bolster the party with an NPC or two
In this solution, you can still play Lost Mines of Phandelver, but you could split the party between you and your girlfriend, with your girlfriend controlling two PCs and you controlling another two (or three if you want to use all five) as NPCs (non player characters - controlled by the DM).
This obviously has the disadvantage of complexity, but it can be fun nevertheless, and gives you and your girlfriend an idea of the different types of characters avaiable to play.
You could also combine 1 & 2 in the following way:
3) Start Phandelver with two (N)PCs, tone things down at the beginning, then add (N)PCs and enemies as you gain confidence
For example, your girlfriend could control one PC, and you one supporting NPC (just choose two from the pregenerated characters).
Spoiler, in case your girlfriend's reading :-)
Cut all encounters in half - so two goblins on the road, half the numbers of goblins in the cave, and either weaken the bugbear in Cragmaw Hideout and/or take his wolf away. Don't forget to halve the XP. Then at the end of Cragmaw Hideout, take Sildar Hallwinter with you and have him stay as a friendly (and tough) NPC. This presumes that your girlfriend will choose to go to Cragmaw Hideout first, but that is the most likely course of action, and you can always guide her back there from Phandalin if she goes there first.
Then as your girlfriend gains confidence playing, she could start to control another PC, and so you will have a tough enough party to face the later more challenging stuff without cutting things down.
For options 1-3 I strongly suggest you have a cleric in your party - parties with clerics last longer!
I haven't tried this myself, but for options 1-3 you might also want to read Play it Solo:Dungeons & Dragons Starter Set. Although it is geared to playing completely solo, looks like it might have some useful advice for converting Phandelver for playing with fewer players.
Or, for something completely different:
4) Find an adventure for one or two players from an earlier edition of D&D and convert it.
This is not my idea, but is described in more detail here:
Solo or 2 player adventures for 5e
You might have trouble doing this conversion just with the basic rules though - you would probably need at least the 5e Monster Manual, though I suppose you could just take the idea (and maps) from such an adventure and swap in monsters from the Starter Set.
Best Answer
Edited on 12/24/2014 to reflect release of the 3 core rule-books.
Starter Set
5th edition's Starter Set is decent to give you a feel for the basic rules, and outside players are not necessary. It is designed with beginners in mind, and is thus a good intro to the game. The starter set has an adventure in it that goes from level 1 to 5, while covering a lot of ground from a role-playing perspective as well as a combat perspective.
It's 20 dollars, but there's a high likelihood that most things outside of using the basic rules to create a character and joining the adventure league as a player are going to cost you more.
It does give you enough to work with to create your own campaigns as well, as the monsters in the appendix are varied enough for some play-room.
Extant Groups
This is particularly advantageous if you don't have a lot of people in your current group and want to get going. If you just combine two groups, you might make the new group too big to manage reasonably. If there's only a couple of you it is easy and would likely be a great intro. If you have a single player you can draw in to play or DM that knows a lot about the game, it can certainly compensate on either side.
Basic Rules
The basic rules are available on the official D&D website. They cover character creation for one of each type of fighter, cleric, rogue, and wizard as well as an assortment of backgrounds. The basic rules are free, and include monsters and magic items in the Dungeon Master documents. They won't tell your DM how to prepare a game outside of managing encounters, but it is enough to get you moving for free if your DM is willing to put in a little work.
Core Rules
As of the time of this update, the three core rule-books are available for purchase at online retailers and local game/hobby-stores.
The Player's Handbook (PHB) covers all of character creation, including far more options than the basic rules and documentation of the rules necessary to play that is more in depth than the basic rules (except in combat, which was covered very well in the basic rules). It is about the price of a new computer game (40-60 USD after-tax depending on where you get them). A game benefits highly from having this book, and most players end up getting one for themselves at some point.
The Monster Manual (MM) and Dungeon Master's Guide (DMG) are more for the DM. The former includes a huge assortment of monsters, beasts, and NPC templates ready for use. The latter is primarily about customizing your campaign, tuning and tweaking it for the way you want to play. The DMG also covers magic items, random dungeon generators, and a host of other resources that can be useful to someone running a game to speed up preparation and allow for customization. Typically only one of each of these books is necessary in an entire gaming group, but with the basic rules for free, the investment can wait until you are ready for it. Each of these two books costs the same as the PHB. Some groups pitch in together to split the cost for these two books, just make sure there won't be problems with book ownership after it's all done.
Summation
If you are in a hurry to try it within the next 3 or 4 weeks, then by all means pick up the Starter Set, or find someone at your local game store that has it and wants to run a game. Ultimately, the decision on how to approach this is up to you. There is no wrong path to start the game.