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You need to use QgsVectorFileWriter.writeAsVectorFormat. If you store one layer per GPKG file you just need:

# get layer list by one of:
#QGIS2
lyrs = iface.mapCanvas().layers() # OR
lyrs = QgsMapLayerRegistry.instance().mapLayers() # OR
#QGIS3
lyrs = iface.layerTreeView().selectedLayers()
lyrs = QgsProject.instance().mapLayers()
lyrs = QgsProject.instance().layerTreeRoot().children()

# select layer
lyr = lyrs[0]
_writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, r"C:\gisData\layer.gpkg")

If you want save more than one layer in gkpg file you should set additional options.

for lyr in filter(lambda l: l.type() == QgsMapLayer.VectorLayer, lyrs):
    options = QgsVectorFileWriter.SaveVectorOptions()
    options.actionOnExistingFile = QgsVectorFileWriter.CreateOrOverwriteLayer 
    options.layerName = "_".join(lyr.name().split(' '))
    _writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, gpkgPath, options)
    if _writer:
        print(lyr.name(), _writer)

PyQGIS should have similar interface for raster layers. In my opinion easiest way to find out is to install IPyConsole plugin and use PyQGIS API

EDIT 02/2020: As @MatthiasKuhn mentioned, to get list of layers in QGIS3 you need to use other methods.

You need to use QgsVectorFileWriter.writeAsVectorFormat. If you store one layer per GPKG file you just need:

QGIS 3:

# imports
from qgis.utils import iface
from qgis.core import QgsProject, QgsVectorFileWriter

# get layer list by one of:
lyrs = iface.layerTreeView().selectedLayers()
lyrs = QgsProject.instance().layerTreeRoot().children()

# select layer
lyr = lyrs[0]
_writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, r"C:\gisData\layer.gpkg")

QGIS 2:

# imports
from qgis.utils import iface
from qgis.core import QgsMapLayerRegistry, QgsVectorFileWriter

# get layer list by one of:
lyrs = iface.mapCanvas().layers() # OR
lyrs = QgsMapLayerRegistry.instance().mapLayers()

# select layer
lyr = lyrs[0]
_writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, r"C:\gisData\layer.gpkg")

If you want save more than one layer in GeoPackage file you should set additional options.

from qgis.core import QgsMapLayer, QgsVectorFileWriter

for lyr in filter(lambda l: l.type() == QgsMapLayer.VectorLayer, lyrs):
    options = QgsVectorFileWriter.SaveVectorOptions()
    options.actionOnExistingFile = QgsVectorFileWriter.CreateOrOverwriteLayer 
    options.layerName = "_".join(lyr.name().split(' '))
    _writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, gpkgPath, options)
    if _writer:
        print(lyr.name(), _writer)

PyQGIS should have similar interface for raster layers. In my opinion easiest way to find out is to install IPyConsole plugin and use PyQGIS API.

You need to use QgsVectorFileWriter.writeAsVectorFormat. If you store one layer per GPKG file you just need:

# get layer list by one of:
#QGIS2
lyrs = iface.mapCanvas().layers() # OR
lyrs = QgsMapLayerRegistry.instance().mapLayers() # OR
#QGIS3
lyrs = iface.layerTreeView().selectedLayers()
lyrs = QgsProject.instance().mapLayers()
lyrs = QgsProject.instance().layerTreeRoot().children()

# select layer
lyr = lyrs[0]
_writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, r"C:\gisData\layer.gpkg")

If you want save more than one layer in gkpg file you should set additional options.

for lyr in filter(lambda l: l.type() == QgsMapLayer.VectorLayer, lyrs):
    options = QgsVectorFileWriter.SaveVectorOptions()
    options.actionOnExistingFile = QgsVectorFileWriter.CreateOrOverwriteLayer 
    options.layerName = "_".join(lyr.name().split(' '))
    _writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, gpkgPath, options)
    if _writer:
        print(lyr.name(), _writer)

PyQGIS should have similar interface for raster layers. In my opinion easiest way to find out is to install IPyConsole plugin and use PyQGIS API

EDIT 02/2020: As @MatthiasKuhn mentioned, to get list of layers in QGIS3 you nedd to use other methods.

You need to use QgsVectorFileWriter.writeAsVectorFormat. If you store one layer per GPKG file you just need:

# get layer list by one of:
#QGIS2
lyrs = iface.mapCanvas().layers() # OR
lyrs = QgsMapLayerRegistry.instance().mapLayers() # OR
#QGIS3
lyrs = iface.layerTreeView().selectedLayers()
lyrs = QgsProject.instance().mapLayers()
lyrs = QgsProject.instance().layerTreeRoot().children()

# select layer
lyr = lyrs[0]
_writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, r"C:\gisData\layer.gpkg")

If you want save more than one layer in gkpg file you should set additional options.

for lyr in filter(lambda l: l.type() == QgsMapLayer.VectorLayer, lyrs):
    options = QgsVectorFileWriter.SaveVectorOptions()
    options.actionOnExistingFile = QgsVectorFileWriter.CreateOrOverwriteLayer 
    options.layerName = "_".join(lyr.name().split(' '))
    _writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, gpkgPath, options)
    if _writer:
        print(lyr.name(), _writer)

PyQGIS should have similar interface for raster layers. In my opinion easiest way to find out is to install IPyConsole plugin and use PyQGIS API

EDIT 02/2020: As @MatthiasKuhn mentioned, to get list of layers in QGIS3 you need to use other methods.

You need to use QgsVectorFileWriter.writeAsVectorFormat. If you store one layer per GPKG file you just need:

# get layer list by one of:
lyrs = iface.mapCanvas().layers() # OR
lyrs = QgsMapLayerRegistry.instance().mapLayers() # OR
lyrs = iface.layerTreeView().selectedLayers()

# select layer
lyr = lyrs[0]
_writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, r"C:\gisData\layer.gpkg")

If you want save more than one layer in gkpg file you should set additional options.

for lyr in filter(lambda l: l.type() == QgsMapLayer.VectorLayer, lyrs):
    options = QgsVectorFileWriter.SaveVectorOptions()
    options.actionOnExistingFile = QgsVectorFileWriter.CreateOrOverwriteLayer 
    options.layerName = "_".join(lyr.name().split(' '))
    _writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, gpkgPath, options)
    if _writer:
        print(lyr.name(), _writer)

PyQGIS should have similar interface for raster layers. In my opinion easiest way to find out is to install IPyConsole plugin and use PyQGIS API

You need to use QgsVectorFileWriter.writeAsVectorFormat. If you store one layer per GPKG file you just need:

# get layer list by one of:
#QGIS2
lyrs = iface.mapCanvas().layers() # OR
lyrs = QgsMapLayerRegistry.instance().mapLayers() # OR
#QGIS3
lyrs = iface.layerTreeView().selectedLayers()
lyrs = QgsProject.instance().mapLayers()
lyrs = QgsProject.instance().layerTreeRoot().children()

# select layer
lyr = lyrs[0]
_writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, r"C:\gisData\layer.gpkg")

If you want save more than one layer in gkpg file you should set additional options.

for lyr in filter(lambda l: l.type() == QgsMapLayer.VectorLayer, lyrs):
    options = QgsVectorFileWriter.SaveVectorOptions()
    options.actionOnExistingFile = QgsVectorFileWriter.CreateOrOverwriteLayer 
    options.layerName = "_".join(lyr.name().split(' '))
    _writer = QgsVectorFileWriter.writeAsVectorFormat(lyr, gpkgPath, options)
    if _writer:
        print(lyr.name(), _writer)

PyQGIS should have similar interface for raster layers. In my opinion easiest way to find out is to install IPyConsole plugin and use PyQGIS API

EDIT 02/2020: As @MatthiasKuhn mentioned, to get list of layers in QGIS3 you nedd to use other methods.