Basically there are few options like 1. worshipping God in fire (Yajna) 2. in idols/images/Yantras etc. These are the most widely used ones.And, these are external worship, where aspirants imagine divine presence outside and separate from themselves.
Apart from these, there is antar-yajna or internal mode of worship, that is suitable for advanced level aspirants and it involves meditative techniques. A simpler version of this mode is called Manasa Puja.
God actually exists everywhere but depending on the person's spiritual level he becomes localised. For the benefit of the aspirant God assumes various forms.
That's why Lord Shiva says:
Agnou Tishtathi ViprAnam Hrudi Devo ManishinAm |
PratimAswapravuddhAnAm Sarvatra ViditAtmanAm ||
For the Brahmins God resides in fire, for the high-souled, he resides
in the heart, for the unintelligent he resides in idols but for the
self-realized he resides everywhere.
KulArnava Tantram 9.44
ChinmayasyAprameyasya nirgunasyA sharirinah |
brahmano rupakalpanA ||
For the benefit of the spiritual aspirants, the conscious (chinmaya),
immeasurable (aprameya), impersonal (nirguna) and bodiless (ashariri)
Brahman has been assigned a form.
KulArnava Tantram 6.73
And, these various forms are fire, idols (which are energized with the help of Mantras).
A deity can be worshipped in various Adharas (object of worship) of which fire, water, idols, Yantras are few.
Linga SthandilaVahnyamvusurpa Kubhyapateshu Cha |
Muddhni Hrudi VA Dasha KirtitA ||
Eshu SthAneshu Devesi Yajanthi
Paramam ShivAm |
ArupAm Rupinim KritvA Karma KAndaratA NarAha ||
Lord Shiva said :
The worshiper visualizes the Formless Devi Parama ShivA in Form and
adores Her in the linga, sign-symbol, altar, Fire, Water, winnowing
fan, wall, image/idol , Mandala (diagram), plank, in the head or in
the heart. These ten places are known to be the different places of
Kularnava Tantram, Chapter 6, Verses 74-75.
Mantra is required in all forms of external worship like Pujas and Yajnas etc - without use of Mantras it is not possible to perform Yajna or Pujas. So, Mantra is not a form, it is a method. A Mantra is actually stated to be non-different from the respective deity.
Different forms of Upasana possible and a hierarchy is also given in the following verse (which is quoted in Yoga Vashishta Sara's 8th chapter ; I don't know the source):
prathamA pratimA-pujA japa-strotAni madhyamA |
uttamA mAnasi pujA
sohaham pujottamottamA ||
The first level of worship is idol-worship, the second level ( which
is of the medium kind) is recitation of mantra/stotra. Manasa puja is
the third level but the highest form is where one worships the God
thinking as not separate from him (Soham-I am HE).