Unearthed Arcana has the classes Druid Variant and Druidic Avenger.
Can I start my character as an Avenger, and later get the other variant?
If it is possible, will my bonus spells be separate? How would I calculate my caster level?
This is explicitly covered in the d20 SRD section on variant classes (and I assume in Unearthed Arcana itself, but my copy is in a box at the moment):
Multiclassing between variants of the same class is a tricky subject. In cases where a single class offers a variety of paths (such as the totem barbarian or the monk fighting styles), the easiest solution is simply to bar multiclassing between different versions of the same class (just as a character can't multiclass between different versions of specialist wizards). For variants that are wholly separate from the character class—such as the bardic sage or the urban ranger—multiclassing, even into multiple variants of the same class, is probably okay. Identical class features should stack if gained from multiple versions of the same class (except for spellcasting, which is always separate).
So in a rules-as-written sense, your DM gets to choose. Personally, I'd say the Druidic Avenger comes under the "different version of the same class", so you wouldn't get to multiclass between them in a game I was running. But I'm not your DM :-)
Unearthed Arcana gives a somewhat-strange suggestion that you might, in some cases, be able to multiclass between two variant classes, taking levels of each. This is somewhat awkward, and personally it’d probably be fairly rare for me to allow it.
On the other hand, what I have always done is allow you to “combine” variants as you like, so long as you’re not trading one feature for two (i.e. can’t take two variants that both lose the same class feature, or at least you only get the replacement class feature from one of the variants). This is more consistent with how alternate or variant class features work than with variant classes, but I find it tends to lead to more reasonable characters.
In all cases, though, bear in mind that variant rules are even more DM-adjudicated than normal.